UV-Finite Gauge Theory and Quantum Gravity

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It is shown that UV-finite Lorentz covariant and gauge invariant quantum field theories can be constructed based on the premise that Lorentzian spacetime is emergent from a statistically averaged …

It is shown that UV-finite Lorentz covariant and gauge invariant quantum field theories can be constructed based on the premise that Lorentzian spacetime is emergent from a statistically averaged ensemble of discrete “pre-spacetimes” having Euclidean signature via a local Wick rotation. A continuum approximation to the underlying discrete theory is developed by expanding fields in terms of a finite number of eigenfunctions of an appropriate gauge covariant Laplacian. The resulting Euclidean spacetime theory is gauge invariant and rotation covariant. It is shown that Wick rotation is uniquely defined locally with respect to a frame at rest relative to a background gravitational field such that rotation covariant results in Euclidean spacetime are mapped to Lorentz covariant results upon Wick rotation. Starting from Hamiltonian formulations, partition functions are derived for Yang-Mills theory and for quantum gravity that contain a finite number of field integrations. A perturbative expansion for the eigenfunctions is developed and it is shown that there are no UV divergences in the theory since there is only a finite number of degrees of freedom. For the case of Yang-Mills theory, quantities that are divergent in the standard approach are replaced with finite quantities that depend on the length scale, L_d, of the discrete pre-spacetimes. In the case of quantum gravity, it is shown that a perturbation expansion in powers of the ratio of the Planck length to L_d can be carried out if it is assumed that this is a small quantity. As concrete examples of the formalism, vacuum polarization and electron self-energy at one loop order in QED are calculated and standard results are obtained with a finite physical cutoff of 2 pi / L_d.

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  • 1. UV-Finite Gauge Theory and Quantum Gravity C. F. Richardson (Dated: February 8, 2014) It is shown that UV-
  • 2. nite Lorentz covariant and gauge invariant quantum
  • 3. eld theories can be constructed based on the premise that Lorentzian spacetime is emergent from a statistically averaged ensemble of discrete pre-spacetimes" having Euclidean signature via a local Wick rotation. A continuum approximation to the underlying discrete theory is developed by expanding
  • 4. elds in terms of a
  • 5. nite number of eigenfunctions of an appropriate gauge covariant Laplacian. The resulting Euclidean spacetime theory is gauge invariant and rotation covariant. It is shown that Wick rotation is uniquely de
  • 6. ned locally with respect to a frame at rest relative to a background gravitational
  • 7. eld such that rotation covariant results in Euclidean spacetime are mapped to Lorentz covariant results upon Wick rotation. Starting from Hamiltonian formulations, partition functions are derived for Yang- Mills theory and for quantum gravity that contain a
  • 8. nite number of
  • 9. eld integrations. A perturbative expansion for the eigenfunctions is developed and it is shown that there are no UV divergences in the theory since there is only a
  • 10. nite number of degrees of freedom. For the case of Yang-Mills theory, quantities that are divergent in the standard approach are replaced with
  • 11. nite quantities that depend on the length scale, Ld, of the discrete pre-spacetimes. In the case of quantum gravity, it is shown that a perturbation expansion in powers of LPl=Ld can be carried out if it is assumed that this is a small quantity. As concrete examples of the formalism, vacuum polarization and electron self-energy at one loop order in QED are calculated and standard results are obtained with a
  • 12. nite physical cuto of 2=Ld. I. INTRODUCTION There are various indications that there may be only a
  • 13. nite number of degrees of freedom associated with any
  • 14. nite volume of spacetime[1{5]. Traditional quantum
  • 15. eld theory associates an in
  • 16. nite number of degrees of freedom with each
  • 17. nite volume of spacetime and it has proven dicult to modify quantum
  • 18. eld theory to have a
  • 19. nite number of degrees of freedom without spoiling Lorentz covariance and/or gauge invariance[6]. Here we construct
  • 20. eld theories having a
  • 21. nite number of degrees of freedom while maintaining both gauge invariance and Lorentz covariance. We do this by considering an emergent spacetime theory that includes a discrete structure and then developing truncated continuous spacetime theories as continuous approximations to the discrete theory. It is well known that one can Wick rotate from a description in Lorentzian spacetime to a de-scription in Euclidean spacetime. The generating functional for the Green's functions in Lorentzian spacetime becomes a partition function in Euclidean spacetime. If we interpret the Euclidean space-time partition function as describing a statistical mechanical system, then it is straightforward to construct a rotationally covariant partition function for a system of extended 4-dimensional objects in Euclidean spacetime - simply include in the de
  • 22. nition of the partition function an average over all orientations of the arrangement of the extended objects. That spacetime or gravity may be related to thermodynamics has been suggested previously; see for example, [7{11]. We consider the extended objects to be components of a pre-spacetime". Lorentzian spacetime may be then be de
  • 23. ned by averaging over all possible arrangements of the extended 4-d objects and then Wick rotating in a speci
  • 24. c reference frame that we will describe below.  This paper is available at http://archive.org/details/DiscreteEmergentSpacetime; and at http://uv
  • 25. nite.weebly.com
  • 26. 2 In this pre-spacetime" picture, quantum gravity can be described as assigning weights to con
  • 27. gurations of the extended objects according to the geometry de
  • 28. ned by each con
  • 29. guration. Quantum
  • 30. eld theory can be described as the statistical mechanics of
  • 31. elds de
  • 32. ned on the con
  • 33. g-urations of the extended objects. Such an approach can be described as an emergent spacetime" approach. That spacetime may be emergent has been previously suggested in various forms; see for example, [12{21]. If we interpret the partition function as de
  • 34. ning a thermodynamic average, this average should be considered a local thermodynamic equilibrium (LTE) approximation. Locally, a frame can always be found where the background spacetime metric is independent of time t up to terms cubic in coordinate displacements and we will assume that LTE holds in this frame[22]. A local Wick rotation can be unambiguously de
  • 35. ned in such a frame. In other words, a background metric speci
  • 36. es a frame in which LTE holds and in this frame a local Wick rotation is uniquely de
  • 37. ned. As we will see in more detail below, this procedure results in a cuto . However, since all observers agree on which frame this cuto is speci
  • 38. ed, Lorentz covariance is maintained. Since partition functions are only determined locally in this approach, we would have to piece local solutions together to determine global physics. Assuming scattering occurs locally, n-point Green's functions could be extended to arbitrary regions of spacetime by determining the inter-acting n-point Green's function in a local region, assuming that the Green's functions are well approximated by combinations of 2-point Green's functions on the boundary of the local region, and then extending the 2-point functions by solving the appropriate wave equation. If we understood the details of the states and the interactions of the extended 4-d objects, we would be able to de
  • 39. ne a partition function by summing over all such states. One approach to doing quantum
  • 40. eld theory in this picture, would be to de
  • 41. ne a lattice gauge theory on an arbitrary lattice, perform standard calculations of correlation functions between
  • 42. elds de
  • 43. ned at two coordinate values (de
  • 44. ning a
  • 45. eld at a coordinate point as the
  • 46. eld on the 4-d object intersecting the coordinate point, for example) and average over all possible lattices. Since such an average includes an average over all orientations, the resulting correlation functions will be SO(4) covariant and give Lorentz covariant Green's functions upon Wick rotation. However, such an approach is dicult to track analytically (beyond simply averaging simple cubic lattices over orientations and displacements) and dicult to extend to the case of quantum gravity without a better understanding of the underlying states. Instead of an approach involving averaging discrete degrees of freedom, we would like to develop continuous approximations to the discrete structures that retain the essential feature of having only a
  • 47. nite number of degrees of freedom in a
  • 48. nite 4-volume of spacetime. To do so, we need to develop a truncated theory in Euclidean spacetime and then Wick rotate the result to Lorentzian spacetime. It is straightforward to maintain Lorentz covariance in the sense discussed above in such an approach - we simply require rotation (i.e., SO(4)) covariance in our truncated theory. It is also necessary to maintain gauge invariance in the truncated theory. We can achieve this by expanding in a truncated set of eigenfunctions of the appropriate gauge (and rotation) covariant Laplacian. Such eigenfunctions are not generally known exactly and we will describe a perturbative approach for their evaluation. Denoting the length scale associated with the underlying discrete pre-spacetime by Ld, the eigenfunctions can be labeled by a 4-vector with a cuto of 2=Ld. (This is, however, not a simple momentum cuto which would not preserve gauge invariance). Note that Ld need not be identi
  • 49. ed with the Planck length. It is possible that the discrete structure characterized by the length scale Ld represents a fundamental building block of nature. Alternatively it is possible that the discrete structure arises as an e ective property of some more fundamental theory. For the purposes of the present paper where we are concerned with truncated continuous approximations, it does not matter whether we consider the discrete structure to be truly fundamental or merely e ective.
  • 50. 3 In the following, we develop truncated rotation and gauge covariant theories generally and derive expressions for the resulting partition function for Yang-Mills theory and for quantum gravity. We argue that such theories are unitary since they can be derived from Hamiltonian formulations and are UV-
  • 51. nite since the
  • 52. elds are expanded in a
  • 53. nite basis. As examples of how to apply the theory, we calculate the vacuum polarization in scalar and spinor QED and the electron self-energy for momenta small compared to 2=Ld in spinor QED. We obtain the standard results with no UV divergences encountered in the calculations. Rather, the cuto appearing in the standard approaches is replaced with the
  • 54. nite physical quantity 2=Ld. For Euclidean quantum gravity there is the well-known conformal mode problem. We will describe the approach of Schleich[23] and argue that this provides an entirely satisfactory solution to the conformal mode problem in our approach. A perturbative expansion for quantum gravity in terms of eigenfunctions of the covariant Laplacian results in a power series in LPl=Ld. If we assume that this ratio of length scales is small compared to unity, then this approach provides a sensible perturbative expansion for quantum gravity that is unitary (since the partition function follows from a Hamiltonian formulation) and that preserves general covariance. Interestingly, it is precisely because quantum gravity is non-renormalizable that such an expansion scheme is possible. This paper is organized as follows. In Section II, we describe a unique local Wick rotation and then in Section III we show how this ties in to Lorentz covariance. In Section IV, we show that expanding in a
  • 55. nite number of eigenfunctions of a gauge covariant Laplacian preserves gauge invariance and we show how to determine the eigenfunctions. In Section V, we illustrate our approach for deriving a functional integral for a system with a
  • 56. nite number of degrees of freedom for a scalar
  • 57. eld and we extend this to the Yang-Mills case in Section VI and to quantum gravity in Section VII. We apply our approach to 1-loop calculations in QED in Section VIII and,
  • 58. nally, we discuss some possible consequences of the theory in Section IX. We use natural units throughout. II. WICK ROTATION As explained in the introduction, it is natural to describe the partition function in quantum
  • 59. eld theory in Euclidean time. In the sections below, we will consider quantum gravity as well as Yang-Mills theory and we will need to be able to de
  • 60. ne a Wick rotation in a curved spacetime. In considering quantum gravity, we would like to interpret the partition function as a statistical average over discrete degrees of freedom. This implies a real metric description in Euclidean time. The Wick rotation to Lorentzian spacetime should also result in a real metric if it is to be physically meaningful. However, it is not possible for a general spacetime to have a real valued metric in the Euclidean time description that is Wick rotated to a real valued metric in Lorentzian time, unless one modi
  • 61. es the de
  • 62. nition of Wick rotation[24, 25]. This diculty is due to the fact that a non-stationary spacetime generally does not represent a system that is in global thermodynamic equilibrium, but local thermodynamic equilibrium could be a reasonable approximation. That thermodynamics of spacetime should be understood locally has been emphasized in [26, 27]. We will now show that the notion of local thermodynamic equilibrium allows us to to de
  • 63. ne a unique local Wick rotation in such a way that both the Lorentzian and Euclidean metrics are real. The metric for an arbitrary stationary spacetime[28] can be written in the form d2s = N(~x)2(dt wi(~x)dxi)2 + (ij + gij(~x))dxidxj (1) where N(~x)2, wi(~x) and gij(~x)) are independent of the time t, and i and j represent spatial coordinates. Wick rotation can be de
  • 64. ned by t ! i and wi ! iwi. Every spacetime is locally stationary. That is around any given point, any Lorentzian metric can be written in the form d2s = dt2(1 + n(~x)) + (ij + gij(~x))dxidxj + dtdxivi(~x) + O(3)dx dx
  • 65. ; (2)
  • 66. 4 where n(~x), gij(~x),and vi(~x) are independent of the time coordinate t,  is the coordinate distance from the reference point which is taken to be ~x = 0; t = 0. To order 2; n, gij , and vi(~x) are quadratic in ~x. Note that equation (2) is of the form of equation (1) with N = 1 + n=2 and wi = vi=2. As we show in Appendix A, this result follows from starting with normal coordinates (g
  • 67. = 
  • 68. ; g
  • 69. ; = 0) at a point and considering coordinate transformations of the form x = x + A
  • 70.  x
  • 71. x x: (3) The coecients A
  • 72.  can be chosen to eliminate all terms linear or quadratic in time. By eliminating time dependence from the metric, we obtain upon Wick rotation, a time-independent metric in Euclidean spacetime and so we can extend the Euclidean time coordinate  so that it ranges from 0 to
  • 73. giving a natural thermodynamic interpretation. The Wick rotated metric is locally d2s = d2(1 + n(~x)) + (ij + gij(~x))dxidxj + ddxivi(~x) + O(3)dx dx
  • 74. ; (4) Note that the term dtdxivi(~x) is the only term in equation (2) which gives a nonzero contribution to R0i which is proportional to T0i by the Einstein equation. Since T0i is proportional to a velocity, it is natural to de
  • 75. ne Wick rotation by t ! i and simultaneously vi ! ivi: This interpretation also follows for Ricci- at spacetime since the twist, which is wi in equation (1), is related to a rotation in the spacetime geometry as is familiar from the Kerr black hole geometry. Our proof (see Appendix A)of equation (2) shows the the coordinate system is uniquely speci
  • 76. ed up to spatial transformations and up to coordinate transformations of the form t = t + Aijkxixjxk: (5) 0i with Aijk 0i 0i = A(ijk). Spatial coordinate transformations do not a ect the de
  • 77. nition of the time coordinate and thus do not a ect the Wick rotation. The local Wick rotation as we have de
  • 78. ned it is a map from a real Lorentzian geometry to a real Euclidean geometry. Transformations of the form of equation (5) shift vas v! v= vjki i i +6Aijkxx. Under Wick rotation, v! iv. This de
  • 79. nes a Wick rotated metric given by equation (4) with vi replaced by v0i. The modi
  • 80. ed Euclidean metric is equivalent (i.e., de
  • 81. nes the same geometry) to the Wick rotated metric determined without making the transformation (5). This can be seen by considering the transformation  =  Aijkxixjxk: (6) which results in the original Wick rotated metric. It follows that our prescription for local Wick rotation in uniquely de
  • 82. ned. This is an intuitively reasonable result: the gravitational
  • 83. eld (i.e., the metric) de
  • 84. nes a local Lorentz frame where the metric is locally stationary. In this frame, the degrees of freedom underlying spacetime are in local thermodynamic equilibrium which can be described in terms of the Wick rotated metric. The prescription for Wick rotation is unique since any boost to the local frame would give a time dependent metric which cannot be directly Wick rotated. This is reasonable since degrees of freedom which are in a thermal distribution in one frame will not have a thermal distribution in a boosted frame. The upper limit of the  integral in de
  • 85. ning a partition function is typically denoted
  • 86. . In order to unambiguously interpret
  • 87. as an inverse temperature, we need it to be a physical distance not just a coordinate distance, because the temperature should not depend on an arbitrary choice of coordinates. If we write the Euclidean metric in the form of equation (4) then the coordinate distance is the physical distance along ~x = 0 and we can unambiguously interpret this distance as the inverse temperature that would be measured by observers at ~x = 0. The inverse temperature for ~x6= 0 is then the physical length of a period for ~x6= 0 and this is given by 1 T(~x) =
  • 88. 1 + n(~x) 1=2 = jg00(~x)j1=2 T(~x = 0) : (7) This implies that T(~x)jg00(~x)j1=2 is constant which is the Tolman-Ehrenfest e ect[29{31].
  • 89. 5 III. LORENTZ COVARIANCE The theories that we discuss below require a cuto . Since we have a unique frame de
  • 90. ned by the metric, all observers agree on which frame this cuto is speci
  • 91. ed and Lorentz covariance is maintained. The Green's functions in our approach will include a manifestly covariant term times a factor like ((2=Ld)2 (k2 + w2)) where k is a wavevector as measured in the unique local frame and w, which on-shell is (k2 +m2)1=2, is the frequency in the same frame. Although at
  • 92. rst sight such a term might appear to violate Lorentz covariance, it is actually a Lorentz scalar. This is because k2 is not the square of a wavevector determined in the frame of an arbitrary observer, but instead is de
  • 93. ned in a way in which all observers agree - it is therefore simply a scalar that does not depend on the reference frame. To illustrate Lorentz covariance in more detail consider a non-interacting scalar
  • 94. eld in an almost- at spacetime background. That is we consider the background spacetime to establish the local frame for Wick rotation but otherwise negligibly di ers from at spacetime. The Euclidean Green's function can be written G(k) = (2=Ld)2 k2   k2 (8) where 2=Ld is a momentum cuto . We will, in general, not be able to truncate the theory with a simple momentum cut-o , but this works for a non-interacting scalar
  • 95. eld since there is no gauge invariance to preserve. Wick rotating gives G(k) = (2=Ld)2 k2 w2   k2 (9) where the k2 in the denominator is now w2 + k2. The k2 appearing in the step function simply records which states prior to Wick rotation gives a non-zero result and as such the w2 term does not get an additional minus sign upon Wick rotation. Now consider a boosted frame where particles with momenta k in the un-boosted frame have momenta ~k in the boosted frame where k =  ~k. The Green's function is G(~k) = (2=Ld)2  P=4 =1( ~k)2  ~k2 = (2=Ld)2 k2 w2   k2 = G(k) (10) where the argument of the step function is determined by the requirement that the modes are cut o in the unique frame where LTE holds and the Wick rotation is valid. Since all observers agree on this frame, the step function does not spoil Lorentz covariance. IV. EXPANSION IN A FINITE BASIS A. Gauge Invariance As mentioned in the introduction, since we must average over discrete con
  • 96. gurations in Eu-clidean spacetime, a possible approach would be to perform a lattice gauge theory calculation for an arbitrary lattice and average the result over all possible lattices. The result of the average is rotation invariant since all possible lattice orientations are averaged over. We therefore obtain a Lorentz covariant result after Wick rotation. While such an approach is faithful to the physical principles that we have described, it is not easily tractable analytically and it is not easily ex-tendable to the case of quantum gravity without additional information on the discrete structures we would be averaging over. We will therefore take an alternative approach were we approximate the discrete structure by a
  • 97. nite number of continuous, spherically symmetric, gauge covariant
  • 98. 6 functions. Doing so allows analytical results to be obtained, but since we are smearing out the discrete structures underlying spacetime, we cannot expect the resulting models to accurately cap-ture physics on an energy scale comparable to the inverse length scale of the discrete structures. Nevertheless, we will be able to obtain the correct low energy physics without UV-divergences appearing since only a
  • 99. nite number of degrees of freedom are included. Our approach will be to expand all
  • 100. elds in terms of eigenfunctions of a gauge covariant Lapla-cian and write functional integrals in terms of the coecients, ak, of this expansion. Truncating the expansion at any given order preserves gauge invariance. To see this, consider Yang-Mills theory. Under a gauge transformation[32] A ! w1Aw + i(@w1)w; (11) the Lie-algebra valued eigenfunctions, i, of the gauge covariant Laplacian DD transform as[32] i ! w1iw (12) where the covariant derivative acting on Lie-algebra valued functions is given by[32] D = @ + i[A; ] (13) and w = ei ata (14) for gauge parameters a and Lie algebra generators ta: (We are using a normalization for the gauge
  • 101. elds such that the coupling constant appears in the trF2 part of the action rather than in the covariant derivative.) We also expand matter
  • 102. elds in terms of the gauge covariant Laplacian i D where the covariant derivative acting on matter
  • 103. elds is given by D = @ + iA. Under a gauge transformation the eigenfunctions, i, of this Laplacian transform as i ! w1i (15) The covariant transformation properties of equations (12) and (15) allow us to preserve gauge invariance when
  • 104. elds are expanded in a
  • 105. nite number of eigenfunctions. To see this, let us expand the vector potential in a
  • 106. nite number of terms as A0  = P i ai0 i and expand in a
  • 107. nite number of terms as = P i ci0i : The superscript 0 indicates that the functions are determined in the absence of a pure gauge term. It is convenient to perform a gauge transformation in order to introduce a pure gauge term that will help us keep track of further gauge transformations. We write A = X i aii + i(@v1)v (16) where i = v10 iv and = X i cii (17) where i = v10i . Under a gauge transformation we have i(@v1)v ! iw1(@v1)vw + i(@w1)w (18)
  • 108. 7 It is readily veri
  • 109. ed that if equations (16) and (17) are substituted into the Yang-Mills action, the action is unchanged under the transformations of equations (12), (15) and (18). It follows that gauge invariance is preserved when the
  • 110. elds are expanded in any
  • 111. nite number of eigenfunctions of the gauge covariant Laplacians. The same conclusion holds in the case of general relativity where we expand the metric in terms of eigenfunctions of  = rara. Under di eomorphisms, tensor eigenfunctions transform as tensors, and similarly for vector and scalar eigenfunctions, and the Ricci scalar constructed from such eigenfunctions transforms as a scalar. However, we generally don't know the eigenfunctions exactly and so they must be determined perturbatively. This generates additional interactions that must be included. We will label the eigenfunctions by a wavevector k and include all eigenfunctions labeled by k  2=Ld with Ld the physical length scale associated with the discrete spacetime structure. B. Perturbation Theory We wish to expand all
  • 112. elds in terms of eigenfunctions of an appropriate gauge covariant Lapla-cian. Since in general we do not know how to determine the eigenfunctions of the full Laplacian, we will need a perturbative expansion. Suppose, for example, that the gauge covariant Laplacian can be written as  = (0) + V and we wish to determine the eigenfunctions as a perturbation expansion in V . In this section, we take the background spacetime to be at and we consider a
  • 113. nite 4-box with width L and apply degenerate perturbation theory to the discrete spectrum. Divide momentum space into thin shells having width on the order of several times 1=L and denote the set of states that diagonalize V in a given shell by jp; ai, where p denotes the momentum mag-nitude and a denotes the additional parameters needed to specify the eigenfunction (i.e., angular variables). Degenerate state perturbation theory gives k;a = jk; ai + X0 p;b jp; bi hp; bjV jk; ai p2 k2 + X0 p;q;b;c jp; bi hp; bjV jq; cihq; cjV jk; ai (p2 k2)(q2 k2) + X0 p;q1;q2;b;c1;c2 jp; bi hp; bjV jq1; c1ihq1; c1jV jq2; c2ihq2; c2jV jk; ai (p2 k2)(q2 1 k2)(q2 2 k2) + ::: (19) where the primes in the sums indicate that terms with any two momenta equal should not be included. (We will generally use primes to indicate that a sum over momenta is restricted in some way. Here, the sums are restricted to exclude poles in the numerator. In other contexts, a prime may be used to indicate a truncated sum.) Standard degenerate state perturbation theory generates terms with two or more momenta equal. However, these terms do not contribute in the thermodynamic limit since such terms contain the same factors of 1=V (4) as the other terms of the same order but contains fewer sums to contribute to limits such as 1=V (4)P k ! R d4k=(2)4 [33]. In equation (19), k;a is not normalized. To leading order the normalization factor is given by N2 k;a  Z d4x k;a(x)k;a(x) = 1 + X0 p;b jhp; bjV jk; aij2 (p2 k2)2 (20) Expanding in this way in principle provides a perturbative expansion of the eigenfunctions and as we have noted, expanding
  • 114. elds in a
  • 115. nite number of such eigenfunctions would provide a
  • 116. nite basis for quantum
  • 117. eld theory that preserves gauge invariance. In particular, a suitable
  • 118. nite basis is k;a with k  2=Ld.[34] However, in practice, we will usually not know how to diagonalize V to obtain jp; ai which is needed to construct k;a. We would therefore like to bypass the problem of diagonalizing V and expand the eigenfunctions in a plane wave basis. We can write jp; ai = X k jk2=p2 Cp;a;k jk i (21)
  • 119. 8 where Cp;a;k are expansion coecients between the two basis sets. Substituting equation (21) into equation (19) and making used of the completeness of states, we see that p;a = X k jk2=p2 Cp;a;k k (22) where k = jk i + X0 p jpi hpjV jki p2 k2 + X0 p;q jpi hpjV jqihqjV jki (p2 k2)(q2 k2) + X0 p;q1;q2 jpi hpjV jq1ihq1jV jq2ihq2jV jki (p2 k2)(q2 1 k2)(q2 2 k2) + ::: (23) For simplicity of notation we have de
  • 120. ned jpi  jp i: The sums in the expansion for k become well de
  • 121. ned principal parts integrals in the large L limit. Since k spans the same set of states as p;a we can use k  k as a basis. This allows us to avoid diagonalizing V , but leads to a normalization factor Nk that does not have a
  • 122. nite large L limit. Since physical results cannot depend on the basis chosen for the expansion and such a divergence would not occur if the p;a basis had been used as this basis gives the correct eigenfunction, physical quantities must either not depend on Nk or the divergence in Nk must cancel with a related divergence. We will explicitly demonstrate below that Nk drops out of the calculation of vacuum polarization in QED at one loop order and that the divergence in Nk cancels with a similar divergence arising from a wave function overlap integral in determining the electron self-energy at one loop order. Note that the functions ~ k  k =Nk with N2 k  R d4x k(x)k(x) satisfy Z d4x~  k0~ k = k0;k (24) and Z d4x~  k0~ k = kk0;k (25) where k = k2 + hkjV jki + ::: (26) Since ~ k(x) satis
  • 123. es these properties of eigenfunctions, we will loosely refer to them as eigenfunc-tions even though it would be more appropriate to reserve that term for the functions p;a(x): V. RESTRICTION TO PHYSICAL STATES To illustrate how the functional integral for quantum
  • 124. eld theory with a
  • 125. nite number of degrees of freedom can be derived, we will
  • 126. rst consider a scalar
  • 127. eld. By discretizing in M spacetime points, where M is
  • 128. nite but much larger than the number of discrete physical degrees of freedom N, our system can be described in terms of an M-dimensional con
  • 129. guration manifold M with an n-dimensional submanifold of physical states N. We derive the partition function by performing a functional integral over the physical submanifold N where the measure is determined as the measure induced on N from the canonical measure on M.
  • 130. 9 We may de
  • 131. ne a physical subspace by restricting (x) to the functions P (x)  X k akk(x; [ak]); (27) where the sum is over N terms and we allow the basis functions to depend on the
  • 132. eld since we will need to do this when we discuss gauge theories. The partition function can be constructed by inserting a set of states that are complete in the physical subspace at Mt time slices. We may discretize in the spatial coordinates as well for a total of M discrete points in the spacetime. We will work in Euclidean time, but it is straight forward to work in Lorentzian time by simply Wick rotating each basis function. Since we need to integrate only over physical states, we can restrict the complete set of states to a set of states that are complete in the physical subspace. This implies that we should integrate only over the physical N dimensional subspace, N, of the full M dimensional space, M. These consideration yield treH
  • 133. = Z dP ()h(xi;
  • 134. )jeHj(xi; j)ih(xi; j)jeH:::eHj(xi; 0)i (28) where dP is the measure on the physical subspace de
  • 135. ned by equation (27) induced by the measure  d on the full M-dimensional space. We will see shortly how to relate dP () to kak. Note that we do not introduce any additional time dependence by restricting the integration to N and so all the time dependence arises from the eH factors which give unitary time evolution upon Wick rotation. As pointed out elsewhere[35], we can obtain a covariant measure if we discretize uniformly in physical distances rather than coordinate distances. So in a curved spacetime, we chose the discretization such that the time slices are p separated by constant Nl where Nl is the lapse and each time slice contains a point in each hx region where h is the determinant of the metric on the constant time slice. We introduce a total of M such points in the spacetime. We now introduce a complete set of momentum eigenstates at each of the M points. One could use a restricted set of momentum states, but it is easier to include complete states at each point since we will exactly integrate out the momenta. A restricted set of momentum states give the same result as the unrestricted set as long as the space spanned by the restricted set includes the classical momentum for each
  • 136. eld con
  • 137. guration (i.e., as long as @ @ _ L can be decomposed in the restricted basis) so that the shift necessary to perform the functional integral can be performed exactly. After integrating over momenta we obtain the standard partition function but integrated over the subspace N of physical states. Now let us determine dP () which can be described as the measure induced on the sub-manifold of physical states N by the volume form on the manifoldMof all
  • 138. eld states. In order to de
  • 139. ne the measure induced on N, we will need to introduce an orthonormal basis for the M N dimensional vector space orthogonal to the N-dimensional tangent space at each point on N. A tangent vector in M can be written @ @i (29) where i are coordinates onMand the subscript i refers to the ith grid point in the discretization. We can de
  • 140. ne an inner product on TpM as  @ @i ; @ @j  = ij (30) where  is the constant 4-volume associated with each point. We may take ak to be coordinates on N. The tangent space in N at a point ak is spanned by @P (xi) @ak @ @i (31)
  • 141. 10 There are N such independent vectors labeled by k. For points close to N, write (xi) = X k akk(xi; [ak]) + X k ck k(xi; [ak]) (32) where k(xi; [ak]) is an orthonormal (in a sense to be de
  • 142. ned below) basis for k > 2=Ld and ck is a corresponding coordinate. Here, the orthonormality condition is determined as follows. A vector in T? p M (the subspace of TpM orthogonal to TpN) can be written @ @ck = X i @ P  p cp p(xi; [ak]) @ck @ @i = X i k(xi; [ak]) @ @i (33) Using the inner product de
  • 143. ned in equation (30), we see that two vectors of the above form are orthonormal, that is  @ @ck ; @ @cp  = k;p; (34) if X i   p(xi; [ak]) k(xi; [ak]) = p;k (35) which is then an appropriate orthonormality condition for k. We also require  @ @ak ; @ @cp  = 0 (36) which implies X i  @P (xi) @ak p(xi; [ak]) = 0 (37) for k  2=Ld and p > 2=Ld. We write the volume measure on M as the product kdakdckJ where J is an appropriate Jacobian factor and then identify dP () as kdakJjck=0: This identi
  • 144. cation follows because @ @ck are unit normal vectors to N. So far we have de
  • 145. ned k(xi; [ak]) only for k > 2=Ld. We now de
  • 146. ne k(xi; [ak]) for k  2=Ld to be an orthonormal basis for the space spanned by @P @ak : The Jacobian, up to irrelevant constant factors, can then be written as X Jjck=0 = det i   k(xi; [ak]) @(xi) @bk0  : (38) where bk0 = ak0 for k0  2=Ld and bk0 = ck0 for k0 > 2=Ld This follows from @p @ak det k;x  = det k;k0 X i Mk(xi) @p(xi) @ak  (39) if detM = 1 (up to constant factors) as it does here. The notation detk;x(:::) refers to a determinant of a matrix labeled by k and x indices. Note that since k is orthonormal, the determinant splits into a component from the k  2=Ld terms times products of unity from the k > 2=Ld terms. Equation (38) can then be rewritten as X Jjck=0 = det i   k(xi; [ak]) @P (xi) @ak0  : (40)
  • 147. 11 where the k; k0 indices in the matrix are now each restricted to be no more than 2=Ld. In practice, we would prefer to avoid having to determine k(xi; [ak]) for k  2=Ld: An alternative approach, which avoids this at the price of having to introduce a second set of ghost
  • 148. elds, is to use @P @ak in place of  k(xi; [ak]) for k  2=Ld. This gives the square of the Jacobian as X J2jck=0 = det i  @P (xi) @ak @P (xi) @ak0 Z  = det d4x p g @P (xi) @ak @P (xi) @ak0  (41) where k and k0 are restricted to be no more than 2=Ld and the right hand side follows in the limit of large M. The Jacobian can then be determined using a combination of Grassmann
  • 149. elds and real scalar
  • 150. elds. Note that equation (41) is a special case of det2A = det(AtBA)=detB which we will use again. The resulting partition function can be written as Z = kdakJjck=0 eS (42) with J given by equation (41). This is covariant since we have discretized in physical distances rather than coordinate distances. If we had quantized by discretizing in constant coordinate lengths rather than physical lengths, we would not obtain a covariant con
  • 151. guration space measure after integrating over momenta. This is because as discussed in [35], the measure would contain spurious factors of the lapse that result from treating degrees of freedom as if they were distributed uniformly in coordinate lengths rather than physical distances. As shown in [35], a spurious factor of the lapse arises for each fermionic degree of freedom and a spurious factor of the inverse lapse arises for each bosonic degree of freedom when the time discretization is uniform in coordinate time rather than physical time. An alternate approach to dealing with these spurious lapse factors is to introduce additional ghost
  • 152. elds to cancel the lapse factors. As an example consider QED or quantum gravity where there are two physical degrees of freedom. The canonical measure determined with uniform coordinate discretization will contain two spurious factors of the lapse. We can eliminate these factors by adding ghost
  • 153. elds having two fermionic degrees of freedom and then integrating out these
  • 154. elds. We will illustrate this approach in section VII. This is a convenient technique for quantum gravity where, as discussed in [35], a direct derivation of the canonical measure by discretizing uniformly in physical time is problematic. VI. YANG-MILLS THEORY A. Derivation of Partition Function Recall that the familiar Faddeev expression for the partition function for a theory with
  • 155. rst class constraints, such as Yang-Mills theory, can be obtained by writing the partition function in terms of physical variables q; p, introducing auxiliary
  • 156. elds Q; as functional integrals over delta functions, and then performing a canonical transformation to return to q; p coordinates[36, 37]. This leaves a delta function for the gauge conditions, a delta function for the constraints, and a Poisson bracket term. The delta function for the constraints and the delta function for the gauge condition can be written in terms of functional integrals over Lagrange multiplier
  • 157. elds c and g, respectively. We may consider q, c and g together to de
  • 158. ne extended con
  • 159. guration variables. As we will see, we can choose gauge conditions and basis functions such that the Fadeev determinant is unity. The resulting partition function is similar to the scalar
  • 160. eld partition function but with more
  • 161. eld variables. We may then truncate the extended variables in much the same way as for the scalar
  • 162. eld case provided that we truncate the extended variables in a way that preserves gauge invariance
  • 163. 12 so that the unphysical auxiliary
  • 164. elds do not contribute to the partition function. To implement this procedure, we de
  • 165. ne a physical" subspace for the gauge
  • 166. eld in terms of a
  • 167. nite set of basis functions that include both gauge
  • 168. xed parts and variations about the gauge
  • 169. xed parts. It is necessary to truncate the gauge variation part as well as the gauge
  • 170. xed part since these variations are components of the extended con
  • 171. guration variables that we wish to truncate. We use a gauge covariant truncation by expanding in gauge covariant eigenfunctions as described previously and we preserve unitarity since the truncation preserves gauge invariance and does not introduce any new time dependence in the functional integral (i.e., the time dependence of correlation functions comes entirely from the eiHt factors). In the approach taken here, the gauge
  • 172. xed part of the gauge
  • 173. eld is explicitly isolated and, as we will see in Section VII, this approach extends directly to the quantum gravity case where we need to explicitly isolate the conformal mode. In addition, our approach allows QED-like Ward identities to be derived for the non-abelian case as we show in the next section. In order to de
  • 174. ne a functional integral over the physical submanifold, we need to be able to split the various components of the functional integral into contributions from the physical subspace and contributions from the orthogonal subspace. We can do this by dividing the expansion of both the physical and unphysical parts of the gauge
  • 175. elds into components in a gauge-
  • 176. xed subspace and components representing variations about this subspace. For the physical part of the gauge
  • 177. eld we denote the basis for the gauge
  • 178. xed part as A(gf)i k : We have Ai(Phys)  = X k2=Ld kA(gf)i (ai k + di kD ik ) (43) where D is the gauge covariant derivative and we denote the expansion coecients for the gauge
  • 179. xed subspace of the physical subspace by ai k and the expansion coecients for the in
  • 180. nitesimal k. ik gauge transformation by di can be taken to be a complete orthonormal set of scalar eigen-functions of the gauge covariant Laplacian, though we only need k  2=Ld in equation (43). The quantity i labels the basis. For simplicity of notation we use the generic label i repeatedly for such a label. The range of i will vary depending on what basis it labels; this should be clear from the context. In order to expand in this way, we need to choose our basis functions such that they contain longitudinal functions of the form Dk. We can do this by taking k(x) to be scalar eigenfunctions of  = DD and de
  • 181. ning a basis for our physical subspace to be functions of the form Dk(x) plus a basis of transverse functions. In general, the vector eigenfunctions of  will not be purely transverse or purely longitudinal since D does not commute with : However, we can form a suitable basis of transverse modes by taking linear combinations of vector eigenfunctions of  with functions of the form Dk(x) to construct functions satisfying DA = 0. We must then take our gauge
  • 182. xing condition such that the set of functions A(gf)i k for k  2=Ld are in the physical subspace de
  • 183. ned by the basis of transverse and longitudinal modes. A natural choice is to require A(gf)i k to be purely transverse, i.e., DA(gf)i k = 0. Denoting the expansion coecients for the orthonormal decomposition of T? p M by ci k we can write A? = X k>2=Ld ci k i k (44) We can rewrite this in terms of a gauge
  • 184. xed (gf) part and a part describing in
  • 185. nitesimal gauge transformations about the gauge
  • 186. xed part as A? = X k>2=Ld (~ci k (gf)i k + ~ di kD
  • 187. ik ) (45)
  • 188. 13 where we denote the expansion coecients for the gauge
  • 189. xed subspace of T? p M by ~ci k, and the p Mby ~ di expansion coecients for the gauge part of T? k. Here
  • 190. ik (x) are chosen to be an independent set of functions such that D
  • 191. ik is orthogonal to Ai(Phys)  . We can use ik and
  • 192. ik to expand an arbitrary gauge parameter as (x) = X k2=Ld di k ik + X k>2=Ld ~ di k
  • 193. ik (46) We take the gauge condition to be 0 = F(x; [A]) = (x) (47) so that the determinant of F(x0)= (x) is unity. We may write the delta function
  • 194. xing the gauge condition as a functional integral over a Lagrange multiplier
  • 195. eld and decompose the Lagrange multiplier
  • 196. eld into a complete set of or-thonormal states. We have xp(F) = Z  i xpd(x)=(2) exp Z d4x p g(x)F(x)  (48) where xp denotes P a product over points uniformly distributed in physical distances[35]. Expanding  as (x) = i k ik (x) where the sum is not restricted to N we have Z d4x p g(x)F(x) = X k2=Ld i kdi k + X k X k0>2=Ld i k ~ dj k0 Z d4x p g
  • 197. k0 k (49) The second term vanishes on N and after restricting the i k integrals to the physical subspace, we have xp(F)jN = ki(di k) (50) Integrating out di k and evaluating on N, we obtain the measure dP (A) = kdaj kJjci k=0;di k=0 (51) By choosing the gauge
  • 198. xing function in this way, we do not obtain additional ghost terms from evaluating the gauge
  • 199. xing determinant. However, the usual Faddeev-Popov ghosts appear in the Jacobian associated with changing variables to ai k; di k. Writing the square of the Jacobian as in equation (41) and introducing a set of Grassmann ghosts and real scalar ghosts to take the square root, we can write the partition function as Z = kdaj kdfik d  fik dbi keSYMSgh (52) where SYM is the Yang-Mills action and the term implementing the Jacobian can be written Sgh = X0 k;k0;i;i0 (  fik fi0 k0 + bi kbi0 k0) X m  @Aj(Phys)  (xm) @^ai k @A(Phys) j (xm) @^ai0 k0  = X0 k;k0;i;i0 (  fik fi0 k0 + bi kbi0 k0) Z p g d4x @Aj(Phys)  (x) @^ai k @A(Phys) j (x) @^ai0 k0  (53)
  • 200. 14 where ^ai k stands for (ai k; di k); the primes indicate that the sums are restricted to k and k0 no more than 2=Ld, and the second form follows in the limit that M ! 1. The j label contains a gauge label as well a basis function label. It is readily veri
  • 201. ed that equation (52) is gauge invariant. In particular, we can change the gauge by replacing the gauge
  • 202. xed
  • 203. eld A(gf)i k with  A(gf)i k where the notation  indicates a gauge transformation for some
  • 204. xed gauge parameter . Such transformations do not alter equation (52). We will make use of this invariance in the next section to derive Ward identities. The partition function of equation (52) can be explicitly evaluated at quadratic order. For each gauge boson, we obtain three factors of Q k1 from the aj k
  • 205. elds and one factor of Q k from the ghost
  • 206. elds, so the partition function is Z = Y0 k2Ng (54) where Ng is the number of gauge bosons. This gives lnZ = Ng X k X n ln(k2 + w2n )((2=Ld)2 (k2 + w2n )) (55) which is the expected form for 2Ng bosonic degrees of freedom. This is a well-de
  • 207. ned partition function that can be evaluated without further regularization. In Appendix B, we show how to exactly evaluate the frequency sum in equation (55) and we show that the standard results for the low energy physics are obtained. At one loop order, calculations based on equation (52) become algebraically much more involved, especially for non-abelian
  • 208. elds. In the present paper, we will content ourselves with evaluating various quantities for QED at one loop order. We will verify that standard results are obtained for QED in Section VIII. Before exploring Ward identities in the next section, let us summarize some properties of the partition function that is evident from our approach. First, perturbation theory will give
  • 209. nite results order by order, since each order of perturbation theory involves a
  • 210. nite number of integrals, each over a
  • 211. nite range of momenta. Second, physical quantities calculated from equation (52) will be gauge invariant since a gauge covariant expansion was used. Third, Lorentz covariance holds as previously discussed. Finally, unitarity holds since the partition function follows from a Hamiltonian formulation where unitarity is manifest. B. Ward Identities In this section, we show that non-abelian Ward identities can be derived for equation (52) that resemble standard Ward identities for QED. First note that matter
  • 212. elds can be introduced in our formalism by adding the appropriate term to the action and expanding P the matter
  • 213. elds in gauge covariant eigenfunctions. Here we consider a fermion
  • 214. eld = ck k belonging to some representation of the gauge group. In our formalism we end up integrating only over a gauge
  • 215. xed subspace rather than an unrestricted integration with an action modi
  • 216. ed by a gauge
  • 217. xing term. We can introduce such a term by considering the action in equation equation (52) to be a functional of A = Ai(Phys)  as given in equation (43), replacing A with  (x)A, expanding the gauge function (x) = P k ga k a k (x) in some basis a k (x), and adding a gauge
  • 218. xing term ga kga k=(2) to the action with a corresponding integral over ga k to the functional integral. Since integration over the ga k
  • 219. elds amount to averaging over gauge equivalent
  • 220. elds, the choice of the basis functions and the number of basis functions is arbitrary. If we choose the basis a k (x) to be independent of ai k; di k; ck; ck and the ghost
  • 221. elds, the action is independent of (x) and we obtain the same partition function as
  • 222. 15 de
  • 223. ned in equation (52), but correlation functions, which are not gauge invariant, will di er from those determined by (52). We can, however, recover correlation functions given by equation (52) by taking the limit  ! 0. To see that the action is independent of (x) when the k are independent of the other
  • 224. elds,
  • 225. rst note that the Yang-Mills part of the action does not depend on such a gauge transformation and so we only need to determine how the ghost part of the action changes. The gauge transformation can be written  (x)A = ei Aei iei @ei (56) In the ghost action, the derivative term gives zero after di erentiating with ^ai k to form Sgh. The derivative only acts on A in the
  • 226. rst term and the factors ei do not contribute to Sgh: This follows because the sum over j in equation (53) includes a sum over gauge
  • 227. eld index and for gauge
  • 228. elds Aa  and Ba  we have the following transformation if we drop the derivative term X a Aa Ba  = 1 C tr AB  ! 1 C tr ei Aei ei Bei  = 1 C tr AB  = X a Aa Ba  (57) where C is a constant. These considerations lead to the generating functional Z[J; ; ] = (kkg k)k2=Ld k adga k daj kdfik d  fik dbi kdckdckeSTOT (58) where STOT = SYM + Sgh + X k ga kga k=(2) + Z d4x JaAa  +  +    : (59) We cuto the ga k terms at k = kg which we can take to be larger than 2=Ld so that equations of the form R d4x (x)f(x) = 0 will imply that f(x) = 0 if f(x) can be decomposed in some
  • 229. nite number of plane wave states as is the case for
  • 230. nite order perturbation theory. Now change variables from ga to ga + a k k k for constant in
  • 231. nitesimal shifts k. The measure does not change and the action changes by STOT = X k ga ka k= + Z d4x JaAa  +  +     (60)  represents the change in Aa  when gk is shifted to gk + k, and similarly for and  . where Aa Since a k is arbitrary we have 0 = X k ga ka k= + Z d4x JaAa  +  +     (61) where h:::i denotes an average under the functional integral. We will take the limit  ! 0, so Aa  is simply D a = D P k a k a k  and the matter
  • 232. elds also follow from the in
  • 233. nitesimal form Pof the gauge transformation. (This is not the case for arbitrary  for a non-abelian theory since k ga k a k (x) does not commute with P k(ga k + a k) a k (x).) Taking a k to be an orthonormal set so that gak = R d4x a(x) a k (x) (no sum on a) and integrating by parts we have 0 = a(x)= DJa + i(ta  ta) (62) Since we are taking the limit  ! 0 our results will hold for calculations based on equation (52).
  • 234. 16 The gauge parameter is a functional of the gauge transformed
  • 235. eld A. [A] can be determined perturbatively in a power series in A: The leading order term can be written [A] = 1@A: (63) It will be sucient to consider only this leading term for deriving relations involving gauge
  • 236. eld propagators and gauge-
  • 237. eld matter vertex functions, but relations among higher order Green's functions require that higher order terms be kept except in the case of QED where equation (63) is exact. Limiting our analysis to propagators and vertex functions, Equation (62) becomes 0 = 1 1@ W Ja  DJajA= W Ja  i(ta W  + W  ta) (64) where W[J] = lnZ[J] and the sign change in the last term comes from commuting Grassmann variables to take the derivative. Following Ryder[38], but using di ering sign conventions since we are working in Euclidean spacetime, we can Legendre transform to form the vertex function , which in our sign convention may be de
  • 238. ned as [A; ;  ] = W[J; ; ] Z d4x JaAa  +  +    (65) We have 0 = 1 1@Aa  D  Aa  ig   ta +  ta     (66) Functionally di erentiating this expression with respect to and then  and setting remaining
  • 239. elds to zero gives @ x1 3   (x3) (x2)Aa (x1) = i  2   (x3) (x2) ta(x2 x1) ta 2   (x3) (x2)  (x3 x1) (67) which can be recognized as a Ward-Takahashi identity relating the derivative of a vertex function to the di erence in inverse propagators. The Ward identity for the polarization can be obtained as follows. First note that evaluating equation (66) with all
  • 240. elds equal to zero gives @  Aa  jA=0 = 0: The only solution to this equation consistent with Lorentz covariance (since there are no 4-vectors around in the absence of
  • 241. elds) is simply  Aa  jA=0 = 0: Next, functionally di erentiate equation (66) with respect to Aa  and then set all
  • 242. elds to zero. In momentum space this gives 0 = 1  k k2 k D 1 (68) where D is the gauge
  • 243. eld propagator (both the context and the number of indices distinguish this from the covariant derivative). Denoting the non-interacting propagator by D(0)ab  = abD(0)  we have D(0)  =  ^k^k k2 + kk (69) which implies D(0)  1 = k2( ^k^k) + 1  ^k^k k2 (70) De
  • 244. ning the polarization  as the di erence between the non-interacting inverse propagator and the interacting inverse propagator, equation (68) then implies k = 0: (71)
  • 245. 17 VII. QUANTUM GRAVITY Quantum gravity can be developed in much the same way as Yang-Mills theory. There are how-ever two signi
  • 246. cant di erences. First, for quantum gravity in Euclidean spacetime, the conformal mode gives a divergent contribution to the functional integral if the action is assumed to be the naively Wick rotated version of the Einstein-Hilbert action. However, the solution to the conformal mode problem suggested by Schleich[23] (similar results were also obtained by others[39, 40]) is a completely satisfactory solution to the conformal mode problem from the perspective of the present theory. We will outline Schleich's argument below. The second major di erence with Yang-Mills theory relates to renormalizability. In the case of Yang-Mills theory, perturbation theory gives expansions in powers of a renormalized coupling constant. In contrast to this, quantum gravity is non-renormalizable. As we show in some detail below, the non-renormalizability allows a perturba-tive expansion to be developed. The quantity LPl=Ld, which we assume to be small, becomes the expansion parameter and the non-renormalizable nature of the theory ensures that higher order loop diagrams gives contributions with higher powers of LPl=Ld so that we have a well de
  • 247. ned perturbative expansion. The essential premise of our approach is that there are only a
  • 248. nite number of physical degrees of freedom associated with a given
  • 249. xed
  • 250. nite 4-volume in Euclidean spacetime. If we allow the metric tensor to vary arbitrarily, we would have uctuations with arbitrarily large 4-volume. We would then not be able to restrict the partition function to a
  • 251. nite number of
  • 252. eld integrations. It is therefore natural to work in the canonical ensemble where the 4-volume is held
  • 253. xed and the metric is expanded in a
  • 254. nite number of modes. We will discuss the grand canonical ensemble elsewhere[41]. A. General Formulation We could expand the metric as g
  • 255. = g(0)
  • 256. + h
  • 257. (72) where g(0)
  • 258. is a background metric. We can decompose h
  • 259. into a transverse, a traceless part, a scalar and a vector part. We may de
  • 260. ne transverse and traceless modes relative to the full metric and write h
  • 261. = TT
  • 262. + 1 4 g
  • 263.  + r( 
  • 264. ) (73) where TT
  • 265. is transverse and traceless,  is a scalar, 
  • 266. is a vector and r is the covariant derivative compatible with the full metric. However, we need to isolate the conformal mode factor. An alternative approach is to
  • 267. rst parameterize a gauge
  • 268. xed metric with 
  • 269. = 0 and then parameterize gauge transformations induced by 
  • 270. : Writing 1 1 4 = e2 we have g(gf)
  • 271. = e2  g(0) + TT
  • 272.  (74) and g
  • 273. =  h g(gf)
  • 274. i : (75) where  is the pullback by the di eomorphism generated by the vector
  • 275. eld  . We may expand the conformal factor in some set of modes i as  = 0 + X i cii: (76)
  • 276. 18 Since we are working in the canonical ensemble, we chose 0 so that the 4-volume inside some
  • 277. xed region[42] is held constant. This makes 0 a function of the other
  • 278. elds. The remaining
  • 279. elds may be expanded as TT
  • 280. = X i aiTT(i)
  • 281. (77) and  = X i dii : (78) In equations (77) and (78), the index i labels both momentum and modes for a given momentum. The appropriate action in Euclidean spacetime is S[g] = S[ci; ai] = SEH[ici; ai]. As discussed in detail by Schleich[23], the Einstein-Hilbert action with an additional factor of i multiplying the conformal factor in Euclidean spacetime gives perturbatively identical results to the Einstein- Hilbert action in Lorentzian spacetime. In Euclidean spacetime this action gives a well de
  • 282. ned functional integral in contrast to the naively Wick rotated Einstein-Hilbert action which does not due to the conformal mode problem. In our action,  or ci represent the conformal mode for a
  • 283. xed gauge slice, which di ers from the de
  • 284. nition of Schleich. However, Schleich's analysis applies directly to . We may write correlation functions with quadratic terms only in the exponent and expand in algebraic powers of the
  • 285. elds outside of the exponent. Then note that if the sign of the part of the Lorentzian action quadratic in ci is reversed and if all factors of ci appearing in algebraic powers outside the exponent are replaced with ici that the expression does not change. Then Wick rotation results in an extra minus sign in the quadratic term in the exponent giving a well behaved conformal mode in Euclidean spacetime. The net result is equivalent to multiplying each ci by a factor of i in the Euclidean action. Note that this conformal rotation" must be applied to all factors of ci, including those associated with ghost terms. We may view Wick rotation as mapping a real Lorentzian metric to a complex metric on Euclidean spacetime of the form ~g
  • 286. =  h e20[ici;ai]+2i [ici; ai] P i cii  g(0) + TT
  • 287. i (79) which can be mapped to the real Euclidean metric of equations (74) and (75) by conformal rotation. From the point of view of a discrete spacetime approach, the discrete structure de
  • 288. nes, at least at low energies, a real Euclidean metric, but the e ective action describing the distribution of metrics is more easily described in terms of the complex
  • 289. eld ~g
  • 290. . As we will see, this approach leads to a partition function that can be written as an integral of eSeff [ici;ai;ghosts] over ai, ci and ghost
  • 291. elds. Although the action here is in a complex form, only real terms contribute to correlation functions. We would obtain equivalent results by de
  • 292. ning  eSeff [ici;ai;ghosts] e^ Seff [ci;ai;ghosts] = Re  : (80) As pointed out by Schleich, the action de
  • 293. ned in this way is not local in the metric g. One might object that a fundamental theory of gravity might be expected to be local in g. However, in our approach g is not a fundamental
  • 294. eld but rather a low energy approximation to discrete degrees of freedom of geometries de
  • 295. ned by the arrangement of the discrete 4-dimensional building blocks of spacetime. So it is not reasonable to require that the action be local in g. Note, however, that although the action is not local in g, it is separately local in  and in TT . To derive the partition function from canonical quantization, we can work directly in Lorentzian spacetime with functions de
  • 296. ned by Wick rotation of basis functions de
  • 297. ned in Euclidean spacetime
  • 298. 19 or we can work in Euclidean spacetime with the complex metric ~g
  • 299. . Alternatively, we can work with the Euclidean metric g
  • 300. and take ci ! ici in the resulting e ective action. We will take this approach here. As shown in [35], canonical quantization gives the measure [g] = Y xp 1 g3 Y  dg( ) det(L
  • 301.  ) = Y xc 1 g3=2 Y  dg( ) det(L
  • 302.  ) (81) where the notation Q xp indicates that the discrete points used in de
  • 303. ning the functional integral are uniformly distributed in physical distances while Q xc indicates uniform discretization in coordinate distances. The equality in equation (81) is to be interpreted as follows. As discussed in [35], the metric can be decomposed in a suitable basis and the both the
  • 304. rst and second forms of the measure can be written in terms of the expansion coecients. The equality in equation (81) means that one obtains the same measure in terms of the expansion coecients starting from either the
  • 305. rst or the second forms. The second form, which allows us to derive the truncated theory in a way that closely parallels Yang-Mills theory, can be derived by introducing ghost
  • 306. elds to cancel the spurious factors of the lapse that would otherwise appear in a uniform coordinate discretization as we now show. Consider the action L = p g g
  • 307. @   @
  • 308.  + m2  (82) where  is a fermionic scalar
  • 309. eld. This would violate the spin-statistics theorem, but we will consider this to be a ghost
  • 310. eld with no physical consequences. We take m >> 2=Ld so there are no propagating modes. By discretizing uniformly in physical lengths as discussed in [35], the partition function for  can be written Z = Z Y xp d(x)d(x) exp  Z d4xL  (83) Expanding  in a basis as  = P k akk with k satisfying R d4x p g kk0 = k;k0 and similarly for , equation (83) becomes Z = Z Y k dakdak exp  Z d4xL   Z Y k dakdak exp  m2 X k akak  (84) where the approximation on the right hand side becomes exact in the limit that m ! 1. Since there are a
  • 311. nite number of terms, this partition function is just a
  • 312. nite (for
  • 313. nite m) constant that does not contribute to any physics. Now consider quantizing the same action and quantize by discretizing uniformly in coordinates. Instead of equation (84) we obtain Z = Z Y k dakdak Y xc N2 exp  Z d4xL  (85) As discussed in [35], the extra factor of N2 appearing in the measure is an artifact of not discretizing uniformly in physical distances. However, if we consider the ghost
  • 314. elds  and  simultaneously with quantum gravity and discretize uniformly in coordinate distances rather than physical dis-tances, the erroneous factors of N cancel. The standard result for the measure in quantum gravity is[43] [g] = Y xc 1 N2g3=2 Y  dg  ( ) det(L
  • 315.  ) (86)
  • 316. 20 The total measure when gravity and the ghost
  • 317. elds are quantized using uniform coordinate dis-tances and the ghost
  • 318. elds subsequently integrated out is given by the right hand side of equation (81). We would like to replace the integral over g
  • 319. with an integral over Ai  (ai; ci; di): To do this,
  • 320. rst de
  • 321. ne an inner product on the space of metrics: (g
  • 322. ; g ) = Z d4x p gg
  • 323. G
  • 324. g  (87) with G
  • 325.  given by the DeWitt supermetric (i.e., metric on the space of metrics) G
  • 326.  = 1 2  g g
  • 327.  + g g
  • 328. + Cg
  • 329. g   (88) where C is any constant greater than 1=2: Since we wish to construct a well de
  • 330. ned ghost action to implement the Jacobian from g
  • 331. variables to Ai variables, we need to require C > 1=2 so that the supermetric is positive de
  • 332. nite. We may simply take C = 0. Up to irrelevant constant factors, we can write the Jacobian from g(xm) variables (with xm distributed uniformly in coordinate distances) to Ai variables as det @g
  • 333. (xn) @Aik = det1=2 hZ d4x p g @g
  • 334. (x) @Ai(k0) G
  • 335.  @g (x) @Ajk i (89) since in 4-d det( p gG) is a constant. If we were using xm distributed in physical distances rather than coordinate distances, we would have an additional factor of detG / g5=2 in this term but we would end up at the same
  • 336. nal result. The Jacobian can be written in terms of ghost
  • 337. elds as det @g
  • 338. (xn) @Ai = Z Y d  fikdfikdbikeSgh[A;] (90) where Sgh = X ijkk0 (  fik0fjk + bi(k0)bjk) Z d4x p g @g
  • 339. (x) @Ai(k0) G
  • 340.  @g (x) @Ajk ; (91) and fi and  fi are fermionic ghost
  • 341. elds while bi is a real scalar ghost. The index i labels transverse-traceless modes for i = 1:::5, the conformal mode for i = 6 and longitudinal modes for i = 7:::10: We will now apply the the steps leading from equation (43) to equation (52). The main dif-ferences here are the additional tensor index on various quantities and how factors of g enter various equations. We will work with uniform coordinate discretization using the right hand side of equation (81). We write g(Phys)  = g(gf)  + X k2=Ld di kr(i )k (92) where g(gf)  is given by equation (74) with sums over modes restricted to k  2=Ld and we denote the expansion coecients for the in
  • 342. nitesimal gauge transformation by di k: In order to expand in this way, we need to choose our basis functions such that they contain longitudinal functions of the form r()k. We can do this by taking k(x) to be vector eigenfunctions of  = rr and de
  • 343. ning a basis for our physical subspace to be functions of the form r()k plus a basis of conformal modes and transverse-traceless functions. In general, the eigenfunctions of  will not be purely transverse or purely longitudinal since r does not commute with : However, we can form
  • 344. 21 a suitable basis of transverse modes by taking linear combinations of tensor eigenfunctions of  with functions of the form r()k to construct functions satisfying rTT  = 0. Instead of using equation (74) for the gauge
  • 345. xed metric, we may take any metric of the form of equation (75) as our gauge
  • 346. xed metric with  a
  • 347. xed element of our basis and with  and TT modes restricted to our basis set. A natural choice for the gauge condition is to simply take the gauge
  • 348. xed metric to be given by equation (74) and to require that the longitudinal modes vanish. Denoting the expansion coecients for the orthonormal decomposition of T? p M by ^ci k we can write g?  = X k>2=Ld ^ci k i k (93) We can rewrite this in terms of a gauge
  • 349. xed (gf) part and a part describing in
  • 350. nitesimal gauge transformations about the gauge
  • 351. xed part as g?  = X k>2=Ld (~ci k (gf)i k + ~ di kr(
  • 352. i )k) (94) where we denote the expansion coecients for the gauge
  • 353. xed subspace of T? p M by ~ci k, and p M by ~ di the expansion coecients for the gauge part of T? k. Here
  • 354. i k(x) are chosen to be an independent set of functions such that r(
  • 355. i )k are orthogonal to g(Phys)  . We can use i k and
  • 356. i k to expand an arbitrary gauge parameter as (x) = X k2=Ld di ki k + X k>2=Ld ~ di k
  • 357. i k (95) We take the gauge condition to be 0 = (x; [g]) = (x) (96) so that the determinant of L
  • 358.  is unity. We may write the delta function
  • 359. xing the gauge condition as a functional integral over a Lagrange multiplier
  • 360. eld and decompose the Lagrange multiplier
  • 361. eld into a complete set of or-thonormal states. We have xc() = Z  i xcg2d(x)=(2) exp Z d4x p g(x)(x)  (97) Expanding  in an orthonormal set as (x) = P i kik (x) where the sum is not restricted to N we have Z d4x p g(x)(x) = X k2=Ld i kdi k + X k X k0>2=Ld i k ~ dj k0 Z d4x p g
  • 362. ik 0i k (98) The second term vanishes on N and we have xc()jN = (xcg3=2)ki(di k) (99) where it has been used that Y xc Y d = ( ki di k) det ( ik (x)) = xc;;ki Y xc Y (g1=2)( ki di k) (100)
  • 363. 22 with the detxc;;ki indicating a determinant between x;  variables and k; i variables with the x points distributed uniformly in coordinate distances. Such determinants were discussed in [35]. Integrating out di k and evaluating on N, we obtain the measure dP (A) = kdaidciJj^ci k=0;di k=0 (101) Using the Jacobian determined in equation (90), the partition function can be written Q = Z Y daikdckd  fjkdfjkdbjkeSEH[a;ic]Sgh[a;ic;f;b] (102) This is our key result for quantum gravity. We denote the partition function for quantum gravity as Q since it can be regarded as a canonical ensemble partition function having a
  • 364. xed number of gravitational degrees of freedom. We will discuss a grand canonical ensemble in [41]. Many properties of equation (102) follow as in the Yang-Mills case. Ward identities can be derived as in Section VIB and in particular it is easy to see that the polarization for the graviton propagator must be transverse. Di eomorphism covariance follows from the covariant expansion that was used and local Lorentz covariance follows from the discussion of Section III. Unitarity fol-lows, as in the Yang-Mills case, from the fact that the starting point was a Hermitian Hamiltonian, no time dependence was introduced in the reduction to the physical subspace, and no dependence on unphysical auxiliary
  • 365. elds was introduced since gauge invariance was preserved. As we will see in the following sections, equation (102) is not only
  • 366. nite, but gives a well de
  • 367. ned perturba-tive expansion in powers of LPl=Ld if this is assumed to be small enough to act as an expansion parameter. Finally, note that even though we have not introduced a cosmological constant into the action, the stationary phase approximation to equation (102) yields Einstein's equation with a cosmological constant. This is because if we consider the action to be a functional of the metric and determine the stationary phase approximation by extremizing the action with respect to variations of the metric, we must introduce a Lagrange multiplier to keep the 4-volume
  • 368. xed. This Lagrange multiplier plays the role of the cosmological constant. We elaborate on the cosmological constant in our formalism elsewhere[41]. B. Quadratic Fluctuations in Quantum Gravity We consider quadratic uctuations in the metric about at spacetime. Neglecting the cosmo-logical constant, the Einstein-Hilbert action to quadratic order in h = g  is given by 16SEH = 1 4 hh + 1 8 hh 1 2 (@h 1 2 @h)2 (103) where h = h. It is sucient to consider the metric to
  • 369. rst order in  and . We have g =  + 2 + TT  + r() (104) where the last term is an in
  • 370. nitesimal gauge term that does not contribute to the Einstein-Hilbert action but does contribute to the ghost action. h is given by 8 in 4 spacetime dimensions. We chose the basis functions for TT  so that the normalization condition Z d4xTT(i)  TT(j)  = ij (105)
  • 371. 23 holds, where i; j label the momenta and the 5 independent basis functions for each momentum. Similarly we take the basis for  to be orthonormal. Equation (103) then becomes 16SEH[a; c] = 6 X k k2ckck + 1 4 X i;k k2ai kai k (106) where i now labels the 5 independent transverse traceless basis functions. So we see that eSEH[a;ic] can be integrated over ai k and ck giving six factors of Q k1. At quadratic level, the ghost part of the action corresponding to ai
  • 372. elds are diagonal and with the normalization condition of equation (105) we have for the transverse traceless contribution to the ghost action STT gh = Xi=5 i=1 X k (  fik fik + bi kbi k): (107) Integrating these ghost terms
  • 373. elds gives a constant factor. Let us divide  into a longitudinal part and a transverse part. The transverse part has three components which we label i for i = 8; 9; 10. (We label these as 8; 9; 10 to match the last three indices in fi and bi as described in the previous section.) These terms give a contribution to the ghost action of ST gh = 1 2 iX=10 i=8 X k k2(  fik fik + bi kbi k) (108) Integrating these ghost
  • 374. elds give three factors of Q k. The remaining components, labeled 6 and 7 and corresponding to the conformal mode and the longitudinal gauge mode, respectively, contribute to a non-diagonal part of the ghost action. The relevant part of the metric can be written as g(67)
  • 375. k = dkk( ^k
  • 376. )eikx + 2ck
  • 377. eikx (109) Denote b6;k the bosonic ghost
  • 378. eld associated with ck and b7;k the bosonic ghost
  • 379. eld associated with dk: Taking C = 0 in equation (88), the ghost action contains b6;k b7;k  16 2k 2k k2  b6;k b7;k  (110) This can be diagonalized if desired, but this is not necessary for our purposes. Note that the eigenvalues are both positive and so the integrals over the ghost
  • 380. elds are well de
  • 381. ned. Integrating Q over the bosonic ghosts and the similar term for the fermionic ghosts gives one factor of k. We have obtained a total of four factors of Q k from the ghost
  • 382. elds and six factors of Q k1 from the six ak and ck
  • 383. elds, so the partition function is Q = Y0 k2 (111) or lnQ = X k X n ln(k2 + w2n )((2=Ld)2 (k2 + w2n )) (112) which is the expected form for 2 bosonic degrees of freedom. This is the same result that we obtained for the Yang-Mills case and it is evaluated in Appendix B.
  • 384. 24 C. Perturbative Expansion in Quantum Gravity The functions , TT
  • 385. and  can be expanded in eigenfunctions of an appropriate covariant Laplacian as described generally in section IV B. An appropriate Laplacian is  = gabrarb. (This is, up to a sign, the Bochner Laplacian. Other Laplacians could be considered, but this is the simplest choice.) 0 is
  • 386. xed by requiring that the 4-volume remain
  • 387. xed. This means that 0 should be expanded in a power series of the other
  • 388. elds. As previously mentioned, a technical diculty arises in constructing basis functions for TT : eigenfunctions of  will not be purely transverse since r does not commute with : However, we can form a suitable basis of transverse modes by taking linear combinations of tensor eigenfunctions of  with functions of the form r()k where k(x) is a vector eigenfunction of . This provides a well de
  • 389. ned
  • 390. nite set of gauge covariant basis functions for expanding the metric. In contrast to Yang-Mills theory, quantum gravity is non-renormalizable. Diagrams that are increasingly divergent in the conventional approach, are proportional to increasingly large powers of the cuto in our approach. Since the cuto is proportional to LPl=Ld; increasingly high or-der diagrams contains factors LPl=Ld to increasingly high powers. If we assume that LPl=Ld is small, then this quantity becomes an expansion parameter that allows a well de
  • 391. ned perturbative expansion to be carried out for quantum gravity. We can explicitly demonstrate the expansion parameter by rescaling as follows. The typical quadratic term in the action can be written in the form 1 L2 Pl X k k2^ak^ak ! V (4) 1 L2 Pl Z d4k=(2)4k2^ak^ak = ~ V Z d4~k=(2)4~k2~a~k~a~k (113) where ~k = Ld=(2)k, ~a~k = ^ak 2 LdLPl , ~ V = V (4) 2=Ld 4 and ^ak represents any of the ai k or ck terms in our expansion of the metric. A term containing n ~ak terms and no ghost terms can be written in the general form  2LPl Ld n2 ~ V n=2 Z d4~k1 (2)4 Z d4~k2 (2)4 ::: Z d4~kn1 (2)4 f(~ki)~a~k1~a~k2:::~a~kn (114) where k~n is minus the sum of the remaining n1 momenta and f(~ki) is a function of the momenta that depends on the term in question. This factor can simply be a dot product of two momenta or it can contain integrable singular terms arising from the perturbative expansion of the eigenfunctions. Ghost
  • 392. elds not in the r( 
  • 393. ) sector can be scaled by a factor of k so that the ghost action is quadratic in k. The ghost
  • 394. elds then scale in the same was as the a^k
  • 395. elds. From the scaling described here, we conclude that 2LPl Ld  serves as an expansion parameter. If Ld is large, we therefore obtain Einstein's equation (with a cosmological constant) as a stationary phase approximation to the partition function with perturbations around the stationary phase solutions giving small contributions. If Ld were not large, it is not clear that Einstein's equation would be meaningful, since uctuations could not be neglected. On the other hand, if Ld were large compared to 103LPl, it would likely cause diculties with any reasonable GUT model. It would seem that a reasonable guess might be that Ld is somewhere in the range of about 10103 Planck lengths. An interesting possibility is that Ld might be on the GUT scale and that gravity could be included with the other forces in a GUT scheme.
  • 396. 25 VIII. QUANTUM ELECTRODYNAMICS A. Scalar QED We expand the scalar
  • 397. eld in eigenfunctions of the gauge covariant Laplacian as (x) = X k ckk(x)=Nk (115) where N2 k = R d4x k(x)k(x) and to second order p V (4)k = eikx + X0 p eipx hpjV jki p2 k2 + X0 p;q eipx hpjV jqihqjV jki (p2 k2)(q2 k2) (116) where V = 2ieA@ + ie(@A) e2AA: (117) The gauge covariant Laplacian for the vector potential is just the ordinary Laplacian and so the vector potential can be expanded as A(x) = 1 p V (4) X k ab kvb keikx (118) where vb k are transverse unit vectors for b=1,2,3, and for b = 4, v4 k is the unit vector parallel or anti-parallel with k. It is convenient to de
  • 398. ne vb k such that vb k = vb (k) so that the reality k) = ab condition on A gives (ab k. We can then apply the techniques of Section VIA except that we need to add matter
  • 399. elds to the partition function. To do this we need the Jacobian of the transformation from f(x) = ((x); (x);A(x)) to bk = (ck; cy k; ai k): Realizing that the complex
  • 400. eld notation is a shorthand for separately integrating over the real and the imaginary parts of the
  • 401. eld and that the Jacobian matrix between Re[], Im[] and ,  is just a constant, the Jacobian can be written, up to irrelevant constant factors, as J2 =  @f @bk ; @f @bk  (119) where the inner product is de
  • 402. ned by (f1; f2) = Z d4x  f1(x)  f2(x) = Z d4x( 1(x)2(x) + 1(x) 2(x) + A1(x)A 2 (x)): (120) Arranging the indicies of the Jacobian matrix with  and  entries ahead of all A entries, we see that the matrix is upper tridiagonal so that it's determinant is the product of the diagonal entries. This product is independent of the
  • 403. elds and so is a constant that can be dropped. The functional measure of equation (51) is then kda1 kda2 kda3 kk where the factor of k arises from the di erence in the Jacobian in equation (51), which is the Jacobian for transforming from A(x) variables to a1 k; a2 k; a3 k; dk = a4 k=k variables, and the constant Jacobian of equation (119). We can introduce a gauge parameter by inserting the factor 1 = Y k Z da4 k(ka4 k Xk)k (121)
  • 404. 26 into the functional integral where Xk is arbitrary. Then following the common procedure, we multiply by e(Xk)2=2 and integrate over Xk for a
  • 405. xed gauge parameter . After introducing the gauge parameter, the functional measure becomes Y dckdc kda k k2e(ka4 k)2=2 (122) where ranges over the three transverse modes and the longitudinal mode and the factor of k2 is needed only to keep track of the appropriate number of degrees of freedom in the partition function (i.e., it cancels 2 of the 4 factors of k1 arising from integrating the 4 a
  • 406. elds). For calculating correlation functions the factor of k2 can be dropped since it is independent of the
  • 407. elds. The scalar
  • 408. eld part of the action can be written S = X k c kck(k[A] m2) (123) where the sign convention that the partition function contains the factor eS is used, k[A] is the eigenvalue of the gauge covariant Laplacian, and the sum should be understood to be restricted to k no more than 2=Ld. The eigenvalue can be written k[A] = k2 X0 p e2Ap  Ap + e2 X0 p ((k + p)  Apk)((k + p)  Akp) p2 k2 : (124) The polarization is then given by ab q = 1 V (4) X0 k 2 k2 + m2  e2 + 4e2 (k + q=2)(k + q=2) (k + q)2 k2  va q vb q (125) where a; b label the basis as in equation (118). By symmetry ab q is proportional to ab for a; b = 1; 2; 3 and 4b q = a4 q = 0 for all a and b. Using i; j to denote the transverse components, we have ij q = 1 3 ij 1 V (4) X0 k 2 k2 + m2  e2 + 4e2 kk (k + q)2 k2  ( ^q^q) (126) or ij q = 1 3 ij 1 V (4) X0 k 2 k2 + m2  3e2 + 4e2k2 1 x2 q2 + 2kqx  (127) where x is the cosine of the angle between k and q. Performing a 4-d spherical average over x, we can write ij q = 1 3 ij 1 V (4) X0 k 2 k2 + m2 2  Z 1 1 dx(1 x2)1=2  3e2 + 4e2k2 1 x2 q2 + 2kqx  : (128) Using the principle parts integral 2  Z 1 1 dx (1 x2)3=2 (q=2k) + x =  q 2k  (3 2(q=2k)2) + 2[(q=2k)2 1]3=2((q=2k) 1); (129) we have ij q = 4e2 3 ij Z k<2=Ld d4k (2)4 1 k2 + m2 h q 2k 2 + 2k q  q 2k 2 1 3=2 i : (130) (q 2k)
  • 409. 27 Using  q = P q va ab q vb q = P ij q vi q vj q, we have  q = 4e2 3 (qq q2) Z k<2=Ld d4k (2)4 1 k2 + m2  1 4k2 2k q3  q 2k 2 1 3=2  (q 2k)  (q2 qq)(q2) (131) which is the form required by gauge invariance and Lorentz covariance. So we have (q2 = 0) = e2 4 3 Z k<2=Ld d4k (2)4 1 k2 + m2 1 4k2 =  1 + (2=(mLd))2 e2 482 ln  (132) and (q2) (q2 = 0) = e2 4 3 Z k<2=Ld d4k (2)4 1 k2 + m2 2k q3  q 2k 3=2 2 1 (q 2k): (133) Substituting z = 2k=q this gives (q2) (q2 = 0) = e2q2 962 Z 1 0 dzz4(z2 1)3=2 1 q2z2=4 + m2 : (134) De
  • 410. ne x = 1 2 [1(1z2)1=2] then z2 = 4x(1x) and [z2 1]1=2dz = 2[z2 1]dx: Using the lower sign converts the integral over z into an integral over x from 0 to 1=2, while using the upper sign converts the integral over z into an integral over x+ from 1=2 to 1. Adding both expressions and dividing by 2, we have (q2) (q2 = 0) = e2q2 962 Z 1 0 dx (1 2x)4 q2x(1 x) + m2 : (135) Using d dx m2 + q2x(1 x) ln  = q2(1 2x) q2x(1 x) + m2 (136) and integrating by parts, we have (q2) (q2 = 0) = e2 42 Z 1 0  dx (x 1=2)2 ln m2 m2 + q2x(1 x)  (137) which agrees with the result obtained in the standard way[44]. We now turn to the more physically interesting case of spinor QED. B. Spinor QED The Euclidean action is S = Z d4x ( (@ + ieA) + m) + 1 4 FF (138) We will refer to the operator i @ as the non-interacting Euclidean Dirac operator. (The factor of i is needed for Hermicity. The factor of 1 is an arbitrary factor that we include to make the ups eigenvectors, which are de
  • 411. ned below, have a positive eigenvalue.) The Euclidean Dirac
  • 412. 28 operator is Hermitian and therefore has real eigenvalues and its eigenvectors form a complete set. To determine the eigenstate, we write the eigenvalue problem as  i@0 i@i i@i i@0  A B  =   A B  (139) where we have used the gamma matrix representation from Sakurai[45]. We substitute upseipx with 4-spinor u to
  • 413. nd the eigenvalue. This implies A = ip p0 B and B = ip p0+ A. Combining these we have  = (p p )1=2. Note that we could have immediately determined the eigenvalues by applying the Dirac operator twice and using the algebra of the gamma matrices. Denoting the eigenvectors with positive eigenvalue as upseipx and the eigenvectors with negative eigenvalues as vpseipx, it is straightforward to verify that ups = N s ip p0+(p p )1=2 s ! (140) and vps = N ip p0+(p p )1=2 s s ! (141) where N2 = p0+(p p )1=2 2(p p )1=2 : From this we obtain X s upsuy ps = p + (p p )1=2 2(p p )1=2 (142) and X s vpsvy ps = p + (p p )1=2 2(p p )1=2 : (143) Note that these equations follow from rotation invariance after being veri
  • 414. ed in a frame where p is in the x0 direction. Note also that uyu and vyv are rotation invariant. There is no need for a factor of 0 to obtain a rotation invariant quantity. This di erence in the Euclidean case from the Lorentz case arises because the generator of rotations S = i 4 [ ; ] is Hermitian in the Euclidean case, but the corresponding generator of Lorentz transformations is not Hermitian because there is no basis for Lorentzian gamma matrices where all gamma matrices are Hermitian. We will drop V0 terms since these can only contribute to tadpole diagrams which vanish by symmetry. Let ksw[A] be the eigenvalue of the interacting Euclidean Dirac operator i (@ + ieA); where k is a momentum index, s is a spin index, and w = 1 denotes the positive or negative eigenvalue. Then to second order ksw[A] = wkjkj + X0 p;wp+k hkswkje = Ap(wp+k(=p + =k) + jp + kj)e = Apjkswki 2jk + pj(wkjkj wp+kjp + kj) ! (144) or ksw[A] = wkjkj e2 X0 p hkswkj = Ap((=p + =k) + wkjkj) = Apjkswki (p + k)2 k2 ! : (145)
  • 415. 29 The fermion and interaction part of the action becomes S = X c kswcksw(iksw + m) (146) and we
  • 416. nd q = e2 1 ab V (4) X0 k;s;wk 2i iwkk + m hkswkj=vb q((=q + =k) + wkjkj)=va q jkswki (q + k)2 k2 ! : (147) Using equations (142) and (143), this can be written q = e2 1 ab V (4) X0 k;wk i (iwkk + m)k tr[=vb q((=q + =k) + wkjkj)=va q (wk=k + k)] (q + k)2 k2 ! : (148) Using the fact that the trace of an odd number of gamma matrices vanish and using tr[=vb q(=q + =k)=va q =k] = 4((k + q)  va q )(k  vb q) + 4(k  va q )((k + q)  vb q) 4(va q  vb q)(k2 + q  k) (149) and tr[=vb q=va q ] = 4(va q  vb q); (150) we have ab q = 0 if either a or b are longitudinal and for the transverse components we have q = e2 1 ij V (4) X0 k;wk 4iwk (iwkk + m)k 2(k  viq )(k  vj q) (viq  vj q)(q  k) (q + k)2 k2 ! (151) or q = e2 1 ij V (4) X0 k 8 k2 + m2 2(k  viq )(k  vj q) (viq  vj q)(q  k) (q + k)2 k2 ! (152) which can be written q = e2 ij ij 3 1 V (4) X0 k 8 k2 + m2 2kk (q  k) (q + k)2 k2 ! ( ^q^q): (153) Letting x be the cosine of the angle between k and q we have q = e2 ij ij 3 1 V (4) X0 k 8 k2 + m2 2k2(1 x2) 3qkx q2 + 2kqx ! (154) or q = e2 ij ij 3 Z k<2=Ld d4k (2)4 4 k2 + m2 2  Z 1 1 dx(1 x2)1=2 (2k=q)(1 x2) 3x (q=2k) + x ! : (155) Using the principal part integral 2  Z 1 1 dx x(1 x2)1=2 (q=2k) + x = (1 2(q=2k)2) + q k  [(q=2k)2 1]1=2((q=2k) 1) (156)
  • 417. 30 and equation (129), we have q = ij e2 ij 3 Z k<2=Ld d4k (2)4 4 k2 + m2   q k 2  3(q=k)[(q=2k)2 1]1=2 + + ((q=2k) 1) 4k=q !  [(q=2k)2 1]3=2 (157) or ij q = e2ij Z k<2=Ld dk 62 k3 k2 + m2 q k 2 2((q=2k) 1)[(q=2k)2 1]1=2 q=k + 2k=q  ! : (158) Writing  q = P ij q viq vj q = (q2)(q2 qq), we have (q2 = 0) = e2 Z k<2=Ld dk 62 k3 k2 + m2 1 k2 = 1 + (2=(mLd))2 e2 122 ln (159) and (q2) (q2 = 0) = e2 Z k<2=Ld dk 32 k3 k2 + m2 1 q2  ((q=2k) 1)[(q=2k)2 1]1=2(q=k + 2k=q)  : (160) Note that since Ld is considered to be a
  • 418. nite physical length scale, (0) is a
  • 419. nite quantity. Making the substitution z = 2k=q we have (q2) (q2 = 0) = e2 Z 1 0 dz 242 q2 q2z2=4 + m2 [z2 1]1=2z2(1 + z2=2): (161) De
  • 420. ne x = 1 2 [1(1z2)1=2] then z2 = 4x(1x) and [z2 1]1=2dz = 2[z2 1]dx: Using the lower sign converts the integral over z into an integral over x from 0 to 1=2, while using the upper sign converts the integral over z into an integral over x+ from 1=2 to 1. Adding both expressions and dividing by 2 we have (q2) (q2 = 0) = e2 Z 1 0 dx 242 q2 q2x(1 x) + m2 (1 2x)2(1 + 2x(1 x)) (162) Using d dx m2 + q2x(1 x) ln  = q2(1 2x) q2x(1 x) + m2 (163) and d dx  (1 2x)(1 + 2x(1 x))  = 12x(1 x) (164) we can integrate by parts to obtain (q2) (q2 = 0) = e2 22 Z 1 0  dx x(1 x) ln m2 m2 + q2x(1 x)  : (165) This is the standard result[44] in Euclidean spacetime with our sign conventions. In contrast to conventional approaches, we obtained this result without encountering any divergences along the way. We still have to renormalize as in the standard approaches, however all counter terms are
  • 421. nite and so the bare charge, for example, is also
  • 422. nite.
  • 423. 31 C. Self Energy The Green's function can be written G(k) = Z d4xd4x0=V (4)eik(xx0) X ws < T ws(x) y ws(x0) > (166) We can split this into the bare propagator plus corrections from the A dependence of the eigen-value, kws, of the un-normalized eigenfunction, kws and of the normalization factor, Nkws as follows G = G(k) G0 = G +  G + NG; (167) where  G = X0 k  ws ^ w^s ~ w~s 1 ik  w + m k ^ w^s < k ^ w^sj(ie = A)jk  ws >< k  wsj(ie = A)yjk ~ w~s > ( ^ wk  wk)( ~ wk  wk) y k ~ w~s (168) G = X ws i(kw kws) (ikw + m)2 kwsy kws (169) and NG = X ws 1 ikw + m N2  kwsy kws 1 kws: (170) Although not explicitly written, the above equations should be understood to include a factor of  (2=Ld)2 q2  for each photon q momenta. Each of equations (169) and (170) also includes a factor of  (2=Ld)2 k2  , while equation (168) includes a factor of  (2=Ld)2 k2  which can be written as (2=Ld)2 k2  (2=Ld)2 k2 =  + h  (2=Ld)2 k2 (2=Ld)2 k2i  : (171) It is readily veri
  • 424. ed that the term in square brackets does not contribute to equation (168) in the limit that k=(2=Ld) ! 0. We will therefore drop this term since our main goal for this section is to verify that the standard result is obtained in the limit that k << 2=Ld. The quantity  (2=Ld)2 k2  is unity in this limit so the only step functions that need to be retained are the (2=Ld)2 q2   factors. In the Feynman gauge, the functional integral over the photon
  • 425. eld gives = Aq ^ w=k + k 2k = Aq ! 1 q2 ^ w=k + 2k k (172) where q = k k. Note that terms proportional to q can be evaluated by replacing q with kk q =k2 since the part of q orthogonal to k integrates to zero in the expressions for G. Using standard properties of the matrices, we have  G + NG = e2 1 V (4) X0 k  ws ^ w^s ~ w~s h( ^ w; ~ w;  w) ( ^ wk  wk)( ~ wk  wk)   ~ w; ^ w ik ^ w + m 1 ik  w + m  (173)
  • 426. 32 where h( ^ w; ~ w;  w) = 1 2 ( ^ w + ~ w)=k=k + ( ^ w ~ w + 1)  1 ^ w  w k 2k  : (174) (1 + k  q=k2) Each of  w; ^ w and ~ w are 1. Keeping track of the discrete number of possibilities, we
  • 427. nd  G + NG = e2 1 V (4) X0 k  w i  w ik  w + m 1 k2 k2 1 k2 + m2 h(k k)(ik  w +m) h+(k + k)(ik  w +m)  (175) where h+ = h(  w;  w;  w) and h = h(  w;  w;  w): Note that both  G and NG contain divergences related to the fact that we determined our eigenfunction perturbative expansion without diagonal-izing in the degenerate subspace. As discussed in Section IV B, such divergences must cancel and we see from equation (175) that indeed they do cancel here. After some algebra, we
  • 428. nd  G + NG = e2 1 V (4) X0 1 (k2 + m2)2 1 k2 + m2 1 q2 + 2k  q    i=k(4(k2 m2) + 4(k2 + k  q) m2 k2 k2  1 q2 : (176) 4m(k2 + k  q + q2) We also have G = 2e2 1 V (4) X0 1 (k2 + m)2 1 q2 + 2k  q i=k(k2 k  q) m2 k2  1 q2 : (177) k2 + 2m(k2 k  q) Combining the separate terms, we
  • 429. nd G = 2e2 1 V (4) X0 1 (k2 + m2)2 1 k2 + m2 i=k(4m2(k2+k q) m2 k2  1 q2 : (178) k2 )+2m(m2+k  q) We will need the following integral which follows from shifting the integration variable: I = Z d4q (2)4 ((2=Ld)2 q2) ((k + q)2 + m2)(q2 + 2) = Z d4~q (2)4 Z 1 0 dz (q2m (x; z) ~q2) (~q2 + 2)2 (179) where 2 = (1z)2k2+(1z)(k2+m2)+z2 and qm(x; z) satis
  • 430. es q2m 2(1z)kqmx+(1z)2k2 = (2=Ld)2 and x is the cosine of the angle between k and ~q. The parameter 2 is a
  • 431. ctitious small photon mass that is introduced to facilitate comparison with standard results. The limit 2 ! 0 is understood. The solution for qm is qm(x; z) = 2 Ld  1 (k(1 z))2(1 x2) (2=Ld)2 1=2 + kx(1 z) 2=Ld  : (180) We have I = 1 83 Z 1 1 dx(1 x2)1=2 Z 1 0 dz  ln q2m (x; z) + 2(z) 2(z)  q2m (x; z) q2m (x; z) + 2(z)  : (181) We will also need J = Z d4q (2)4 ((2=Ld)2 q2) ((k + q)2 + m2)(q2 + 2) k  q k2 = Z d4~q (2)4 Z 1 0 dz (q2m (x; z) ~q2) (~q2 + 2)2 (k~qx=k2 (1 z)) (182)
  • 432. 33 or J = 1 43k Z 1 1 dx(1 x2)1=2x Z 1 0 dz  qm 3 2 tan1(qm=) + qm 2 2 2 + q2m  I2 (183) where I2 is the integral for I with a factor of (1 z) inserted into the integrand. So we have G = 2e2 1 (k2 + m2)2 i=k (I + J)(m2 k2) 4m2I  + 2m(m2I + k2J): (184) From this the self energy is determined to be  = 2e2 i=k(I + J) + 2mI  : (185) We have already assumed that k << 2=Ld in dropping the term in square brackets in equation (171). Taking m << 2=Ld as well and expanding accordingly, we
  • 433. nd that to leading order  = e2 82 Z 1 0 dz 2m + iz=k   ln z(2=Ld)2 (1 z)m2 + z2 + z(1 z)k2  : (186) So we recover the standard result[44] for momentum and mass small compared to 2=Ld. There are corrections to continuum spacetime quantum
  • 434. eld theory, however, when the momentum is not small compared to 2=Ld. IX. DISCUSSION AND CONCLUSION We started from the notion that underlying the Lorentzian spacetime that we experience is a discrete pre-spacetime that must be averaged in a statistical mechanical sense to determine the Lorentzian spacetime. The discrete structure in this picture is considered to consist of extended 4- dimensional objects that make up the pre-spacetime and de
  • 435. ne a metric with Euclidean signature. Quantum
  • 436. eld theory in our approach can be described as the statistical mechanics of
  • 437. elds since the Euclidean domain is considered to be fundamental. Lorentzian time is not fundamental, but rather is an emergent quantity that is de
  • 438. ned by the Wick rotation procedure. We can think of time in this picture as measuring the evolution of a system that is perturbed from equilibrium. We developed a continuous approximation to the discrete pre-spacetime structure where the cuto is the only feature of the discrete structure that is retained. Such an approach gives a well de
  • 439. ned description of low energy physics without any UV divergences. An alternate approach, which we did not pursue here, would be to work directly in the discrete Euclidean pre-spacetime. One can, in principle, conduct a lattice gauge theory calculation for each lattice" (not necessarily periodic) in an ensemble of lattices and average the results. This will give an SO(4) covariant result in Euclidean spacetime and will generate a Lorentz covariant result after Wick rotation. The main disadvantage of such an approach for gauge theories is that it is not analytically tractable unless one restricts the lattices to simple tractable lattices. For quantum gravity, the main diculty is that we don't know what the discrete action actually is or even precisely what the relevant degrees of freedom are at a fundamental level, though according to our approach, the relevant degrees of freedom must relate to the con
  • 440. gurations of the discrete extended 4-dimensional building blocks of spacetime. As long as we are interested in low energy physics, however, a continuous approximation to the discrete structure is reasonable. We have shown that UV-
  • 441. nite gauge theories can be constructed by expanding
  • 442. elds in a
  • 443. nite set of gauge covariant basis functions. We have derived partition functions for Yang-Mills theory and for quantum general relativity using this approach. Let us summarize the properties
  • 444. 34 of these partition functions. First, the partition functions are
  • 445. nite since they involve a
  • 446. nite number of integrals which are well de
  • 447. ned, at least perturbatively, since the quadratic part of the action is positive. Second, the partition functions generate unitary evolution since each partition function follows from a Hamiltonian where unitarity is manifest. In the case of Yang-Mills theory, renormalization can be carried out as is usually the case except that divergent quantities in the conventional approaches are replaced with quantities such as 2=Ld which would diverge if Ld were taken to zero. To verify our formalism, we have presented detailed calculations for vacuum polarization and electron self-energy in QED and we have obtained standard results for the
  • 448. nite parts of these quantities with no UV-divergences arising in our approach. In the case of quantum gravity, the partition function generates an expansion in powers of 2=Ld: If this is assumed to be a small quantity, we obtain a well de
  • 449. ned perturbative expansion for quantum gravity that preserves general covariance and, according to the derivation of the partition function, generates unitary time evolution. Although detailed calculations for quantum gravity beyond the level of quadratic uctuations remain to be worked out, our approach would appear to provide a promising framework for understanding quantum gravity. The partition function for quantum gravity yields Einstein's equation with a cosmological con-stant as a stationary phase approximation. The cosmological constant enters as a Lagrange mul-tiplier which must be introduced in order to vary the metric while holding the 4-volume
  • 450. xed. It is necessary to hold the 4-volume
  • 451. xed when varying the metric since the theory is formulated in terms of a
  • 452. nite number of degrees of freedom for any given 4-volume. We have limited our discussion in the present paper to the basic formulation of the theory and some basic calculations in QED. We will discuss in more detail elsewhere[41], the relationship between the Lagrange mul-tiplier, the chemical potential and the cosmological constant. As discussed in [41], the existence of a cosmological constant with a magnitude many orders of magnitude below the Planck scale can be interpreted as an indication that discrete degrees of freedom underly spacetime. We will also discuss the application of the discrete spacetime approach to black holes and the early universe elsewhere[46] and we will argue following [5] that a spectral index observed to be about 0.96 can be interpreted as an indication that discrete degrees of freedom underly spacetime. We have emphasized that it is necessary to understand quantum
  • 453. eld theory and quantum gravity in terms of a local thermodynamic equilibrium approximation. Although we have dis-cussed only local physics, non-local physics can in principle be determined by piecing together neighboring regions. For example, although we have de
  • 454. ned the Green's function in terms of a single region in local thermodynamic equilibrium, we can in principle piece together many such regions to construct a Green's function describing propagation over arbitrarily large distances. This can be approximately achieved by modeling interactions locally and extending the 2-point functions to arbitrary distances by solving the generally covariant di erential equation satis
  • 455. ed by the propagators. Our approach provides a sensible perturbative expansion for quantum gravity if the length scale Ld is large compared to the Planck scale. Since the GUT scale is only a few orders of magnitude below the Planck scale, an interesting possibility that Ld could be at or near the GUT scale. This would open the possibility that any uni
  • 456. ed theory should be based on a discrete underlying spacetime rather than a continuous spacetime as is typically assumed and also suggests that gravity may need to be included with the other three forces in a GUT scheme. Another interesting possible consequence of having Ld near the GUT scale is a modi
  • 457. cation to the ow of the coupling constants as the GUT scale is approached. As is well known, the strong, weak and QED coupling constants almost but don't quite meet at the GUT scale according to conventional calculations. This is typically considered to be indirect evidence for supersymmetry which can modify the coupling constant ow so that the three coupling constants meet at a common scale. The renormalization group calculations which determine the coupling constant ow are based on a continuous spacetime
  • 458. 35 description. An interesting possibility is that the coupling constant ow is modi
  • 459. ed by discrete spacetime e ects as the energy scale approaches the GUT scale. This does not occur for the QED coupling in the calculations that we have presented here at one loop order. However, these calculations are based on a continuous description of a discrete structure where the cuto is the only feature of the discrete structure that is retained. One would expect corrections that are not included in such a continuous description could become important for energy scales on the order of 1=Ld. If discrete spacetime e ects modify the ow of the coupling constants as the GUT scale is approached, it is possible that supersymmetry enters at a higher energy scale so that a smaller shift in the coupling constant ow is needed or it is possible that supersymmetry is not needed at all for the coupling constants to meet at a common energy scale. Appendix A: Local Frame for Wick Rotation Here we show that locally about any point any Lorentzian metric can be put in the form of equation (2). We start with normal coordinates (for example, Riemann normal coordinates) and consider a series of coordinate transformations of the form x = x + A
  • 460.  x
  • 461. x x: (A1) We will keep only terms up to quadratic order in coordinate distances and we show that terms not included in equation (2) can be eliminated with a suitable choice of A
  • 462. . The proof proceeds in several steps in which separate terms containing one or two powers of time are eliminated. Some steps generate additional time-dependent terms that are eliminated in subsequent steps. Each step can otherwise be treated independently since we are dropping cubic and higher order terms. We generically denote coecients of the coordinate changes by A (with appropriate superscripts and subscripts) in each step, but the A's in each step are independent. 1. Elimination of dxidxj t2 and dtdx[ixj] t terms Consider xi = xi + t2xjAi j (A2) The metric picks up a term of the form 4dt dxi xj t Aij + 2t2 dxi dxj Aij (A3) where indicies on Aij are raised and lowered with ij . A(ij) can be chosen to eliminate the dxidxjt2 terms and A[ij] can be chosen to eliminate dtdx[ixj]t terms. 2. Elimination of dtdx(ixj)t terms Consider t = t + txixjAij (A4) The metric picks up an x-dependent dt2 term that we don't need to eliminate and a term of the form 4dtdx(ixj)tAij (A5) Aij can be chosen to eliminate the dtdx(ixj)t terms. 3. Elimination of dt2 t2 terms Consider t = t + t3A (A6)
  • 463. 36 The metric picks up a term proportional to dt2 t2 A and so A can be chosen to eliminate the dt2 t2 terms in the metric. 4. Elimination of dt2 t x terms Consider t = t + t2xiAi (A7) This gives a term proportional to dt2 t xiAi which can be used to eliminate dt2 t x terms in the metric. It also generates a dt dxiAi t2 term which will be eliminated in the next section. 5. Elimination of dt dxi t2 terms Transformations of the form xi = xi + t3Ai can be used to eliminate dt dxi t2 terms. 6. Elimination of Ckij t xk dxi dxj terms Here Ckij = Ck(ij) is a set of constants. First note the following Ckijxkdxidxj = Ckijx(kdxi)dxj + 1 2 Ckijxkdxidxj 1 2 Ckijxidxkdxj : (A8) Since Ckij = Ck(ij), the last term can be rewritten as 1 2Ckijx(idxj)dxk: Relabeling indicies in this term and moving the second term to the left hand side we have Ckijxkdxidxj = 2C(ki)j Cj(ki)  x(kdxi)dxj (A9) The coordinate transformation xi = xi + txj xkAi jk (A10) generates the metric term 4tx(kdxj)dxiAijk + 2dtdxixjxkAijk (A11) The second term will be addressed in the next section. The
  • 464. rst term can be used to eliminate the Ckijt xk dxi dxj term with the choice Ajki = 1 4 2C(ki)j Cj(ki)  (A12) Steps 1-6 puts the metric in the form of equation (2). We cannot eliminate a term in the metric of the form dtdxixjxkBijk where Bijk is a set of constants since this term contains physically meaningful information on rotation in the spacetime geometry. We can however eliminate a part of this term. 7. Elimination of symmetric part of dtdxixjxkBijk We can keep a term like dtdxixjxkBijk in the metric and still have a real metric in both Lorentzian and Euclidean spacetime by requiring Bijk ! iBijk under Wick rotation. Such a term generates an R0i term which is proportional to a velocity by the Einstein equation, so it is sensible to require Bijk to pick up a factor of i under Wick rotation. However, the completely symmetric part of Bijk can be eliminated by a coordinate transformation of the form t = t + xixj xkAijk: (A13) This generates a term in the metric proportional to dtdx(ixj xk)Aijk (A14) The coecients Aijk can be chosen to set B(ijk) = 0: Note that Aijk is completely determined by the condition B(ijk) = 0; so if we impose B(ijk) = 0 as a gauge condition, the metric of equation (2) is uniquely determined up to transformations of the spatial coordinates.
  • 465. 37 Appendix B: Exact Summation for the Partition Function for Yang-Mills Theory and for Quantum Gravity at Quadratic Order In this appendix, we derive an expression for the sum over frequencies appearing in equations (55) and (112). The partition function for quantum gravity can be written as lnQ = X0 k nX=m n=m ln(k2 + w2n ) (B1) where wn = 2n=
  • 466. and m, which is the integer part of ((2=Ld)2 k2)1=2
  • 467. =(2), is the maximum value of n where the step function is satis
  • 468. ed. The prime on the sum over k indicates that the sum is restricted to k2  (2=Ld)2. The partition function is a
  • 469. nite sum of
  • 470. nite terms and is therefore obviously
  • 471. nite and so regularization is not necessary in order to evaluate Equation (B1). However, zeta function regularization allows us to derive the standard results for the partition function from Equation (B1) up to small corrections due to the cuto . We de
  • 472. ne the generalized zeta function as (s) = X0 k nX=m n=m s k;n: (B2) Note that this is a sum of a
  • 473. nite number of
  • 474. nite analytic functions of s and so (s) is an analytic function of s. The partition function can be written lnQ = 0(0): (B3) We now add and subtract an in
  • 475. nite number of terms to obtain (s) = X0 k nX=1 n=1 s k;n 2 X0 k nX=1 n=m+1 s k;n: (B4) The
  • 476. rst term is the usual zeta function for a boson
  • 477. eld and thus gives the usual partition function in Equation (B3). The second term gives two types of corrections: 1) the zero point energy is shifted, and 2) there are additional terms that vanish in the limit Ldk ! 0. To see this, write the second term as ^(s) = X0 k ^k(s) = 2 X0 k nX=1 n=m+1 s k;n = X0 k 1 (s) Z 1 0 ts1Y (t)dt (B5) where Y (t) is given by Y (t) = 2 nX=1 n=m+1 exp(k;nt) = 2 nX=1 n=m exp((k2 + (2n=
  • 478. )2)t): (B6) By shifting the origin of n this is Y (t) = 2 nX=1 n=1 exp((k2 + (2(n + m)=
  • 479. )2)t) (B7) which gives 1 2 (s)^k(s) = nX=1 n=1 Z 1 0 ts1 exp((k2 + (2(n + m)=
  • 480. )2)t)dt: (B8)
  • 481. 38 Writing  = (n+m)2
  • 482. 2 t and expanding the k2 part of the exponent, we have 1 2 (s)^k(s) = 1X n=1 Z 1 0 d
  • 483. 2s 1X l=0 (
  • 484. k)2l(1)l s+l1 (n + m)2l+2sl! e42 : (B9) Carrying out the integral and changing the order of summation (we can always choose s to be in a region of the complex plane where this is valid and obtain the result for other s via analytic continuation), we have 1 2 (s)^k(s) = 1X l=0 1X n=1
  • 485. 2s (
  • 486. k)2l(1)l(s + l) (n + m)2l+2sl!(2)2s+2l : (B10) Noting that 1X n=1 1 (n + m)2l+2s = (2l + 2s;m + 1) (B11) where (s; q) = P1 n=0(q + n)s is the Hurwitz zeta function, we have 1 2 (s)^k(s) = (
  • 487. 2 )2s(s)(2s;m + 1) + 1X l=1
  • 488. 2s (
  • 489. k)2l(1)l(s + l) l!(2)2s+2l (2l + 2s;m + 1): (B12) This gives 0 k(0) = 2 ^ @ @s  (2s;m + 1)(
  • 490. 2 )2s  s=0 + 2 1X l=1 (
  • 491. k)2l(1)l l(2)2l (2l;m + 1): (B13) Using the Leurch identity (Gradshteyn and Ryzhik [47] 9.533-3)  0 (0;m + 1) = ln (m + 1) 1 2 ln(2) = ln (m) 1 2 ln(2) + ln(m) (B14) and 1X l=1 (
  • 492. k)2l(1)l l(2)2l (2l;m + 1) = 2 ln (m + 1) + ln (m + 1 i
  • 493. k=(2)) + ln (m + 1 i
  • 494. k=(2)) = 2 ln j(m + i
  • 495. k=(2))=(m)j + ln(1 + (
  • 496. k=(2m))2); (B15) where the last expression is valid for
  • 497. k real, we obtain 0 k(0) =4 ln (m) 2 ln(2) 4(m + 1=2) ln(
  • 498. =2)+ ^ 4 ln j(m + i
  • 499. k=(2))=(m)j + 2 ln(1 + (
  • 500. k=(2m))2) + 4 ln(m): (B16) We can write this as 0 k(0) =4 ^  ln (m) (m 1=2) ln(m) + m 1 2  ln(2) 4(m 1=2) ln(
  • 501. =(2m)) + 4 ln j(m + i
  • 502. k=(2))=(m)j + 2 ln(1 + (
  • 503. k=(2m))2) + 4 ln(m) 4 ln(
  • 504. =2) 4m: (B17)
  • 505. 39 Using ln(1 + (
  • 506. k=(2m))2) ln((
  • 507. =(2m))2) = ln((2=Ld)2), this becomes 0 k(0) =ln((2=Ld)2) 4m(1 + ln(
  • 508. =(2m))) + 4 ^  ln (m) (m 1=2) ln(m) + m 1 2 ln(2)  + 4 ln j(m + i
  • 509. k=(2))=(m)j + ln(1 + (
  • 510. k=(2m))2): (B18) The partition function is then given by lnQ = X0 k 
  • 511. k + 2 ln(1 e
  • 512. k) + ^  : (B19) 0 k(0) The
  • 513. rst term in equation (B18) is an irrelevant constant that can be eliminated by symmetrically keeping 1/2 of the boundary terms in the sum. The second term is proportional to
  • 514. and therefore represents a shift in the zero point energy. The remaining terms vanish in the limit Ld ! 0 for
  • 515. xed p and therefore do not contribute to the low energy physics. Thus the conventional Riemann zeta function regularization gets the low energy physics correct. Note that we cannot assume that the high energy physics determined by the partition function is physically meaningful since, as discussed in Section IV, we are smearing out the discrete structure by using a
  • 516. nite basis of continuous functions rather than averaging over discrete structures and therefore equation (B19) should not be taken as a physically meaningful result for k approaching 2=Ld. Note that the partition function given in equation (B1) is also the partition function for a non-interacting complex massless scalar
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  • 518. 40 [22] There are two senses in which the word local" may be used. Throughout most of this work we use the term local" to refer to the local frame in which Wick rotation is de
  • 519. ned. This frame has some
  • 520. nite extent. That is, there is some
  • 521. nite length scale in which the metric is well approximated by the form discussed in Section II. The word local" may be used in a narrower sense when discussing interactions where local" means at a common point. For example, (x)(x) is a local interaction while (x)K(xy)(y) is not. Non-local e ects in the latter sense could be local in the former sense. It should be clear from the context which notion of local" is intended. [23] K. Schleich, Phys. Rev. D 36, 2342 (1987). [24] A. Dasgupta and R. Loll, Nucl. Phys. B 606, 357 (2001), arXiv:hep-th/0103186v2. [25] A. Dasgupta, JHEP 0207, 062 (2002), arXiv:hep-th/0202018v3. [26] T. Padmanabhan, A dialog on the Nature of Gravity," (2009), arXiv:0910.0839v2. [27] T. Padmanabhan, A Physical Interpretation of the Gravitational Field Equations," (2009), Based on the Plenary talk given at the International Conference on 'Invisible Universe', 29 June - 3 July, 2009, Paris, arXiv:0911.1403. [28] R. M. Wald, General Relativity (The University of Chicago Press, 1984). [29] R. C. Tolman, Phys. Rev. 35, 904 (1930). [30] R. C. Tolman and P. Ehrenfest, Phys. Rev. 36, 1791 (1930). [31] C. Rovelli and M. Smerlak, Thermal time and the Tolman-Ehrenfest e ect: temperature as the speed of time," (2010), arXiv:1005.2985v3. [32] C. Cronstrom, On the uniqueness of solutions to the gauge covariant Poisson equations with compact Lie algebras," (2005), arXiv:hep-th/0502102v1. [33] An additional factor of the length L can arise from summing such terms over close eigenvalues (i.e., a momenta di erence on the order of 1=L). However, the factor of 1=V (4) goes to zero fast enough as L ! 1 that terms with two or more momenta equal do not contribute. [34] Note that the momenta sums in equation (19) are unrestricted. When a
  • 522. nite basis is used, however, each sum terminates at some
  • 523. nite momenta which will generally not be less than 2=Ld. [35] C. F. Richardson, Manifestly Covariant Functional Measures for Quantum Field Theory in Curved Spacetime via Canonical Quantization," (2014), archive.org/details/CovariantFunctionalMeasure. [36] R. Marnelius, Acta Physica Polnica B13, 669 (1982). [37] H. J. Rothe and K. D. Rothe, Classical and Quantum Dynamics of Constrained Hamiltonian Systems (World Scienti
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  • 526. ne the region to be a 4-cube of length L where L is to be taken to in
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