Part IV - Quantum Fields

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PART IV - Continuation of PART III - Quantum Mechanics.

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Part IV - Quantum Fields

  1. 1. From First Principles PART IV – QUANTUM FIELDS March 2017 – R3.0 Maurice R. TREMBLAY
  2. 2. “A poet once said, ‘The whole universe is in a glass of wine.’ We will probably never know in what sense he meant that, for poets do not write to be understood. But it is true that if we look at a glass of wine closely enough we see the entire universe. There are the things of physics: the twisting liquid which evaporates depending on the wind and weather, the reflections in the glass, and our imagination adds the atoms. The glass is a distillation of the earth’s rocks, and in its composition we see the secrets of the universe’s age, and the evolution of stars. What strange array of chemicals are in the wine? How did they come to be? There are the ferments, the enzymes, the substrates, and the products. There in wine is found the great generalization: all life is fermentation. Nobody can discover the chemistry of wine without discovering, as did Louis Pasteur, the cause of much disease. How vivid is the claret, pressing its existence into the consciousness that watches it! If our small minds, for some convenience, divide this glass of wine, this universe, into parts – physics, biology, geology, astronomy, psychology, and so on – remember that nature does not know it! So let us put it all back together, not forgetting ultimately what it is for. Let it give us one more final pleasure: drink it and forget it all! ” Richard Feynman Epicatechin TARTARIC ACID (C4H6O6) 2,3-dihydroxybutanedioic acid Tartaric acid is, from a winemaking perspective, the most important in wine due to the prominent role it plays in maintaining the chemical stability of the wine and its color and finally in influencing the taste of the finished wine. [Wikipedia] MALIC ACID (C4H6O5) hydroxybutanedioic acid Malic acid, along with tartaric acid, is one of the principal organic acids found in wine grapes. In the grape vine, malic acid is involved in several processes which are essential for the health and sustainability of the vine. [Wikipedia] CITRIC ACID (C6H8O7) 2-hydroxypropane-1,2,3-tricarboxylic acid The citric acid most commonly found in wine is commercially produced acid supplements derived from fermenting sucrose solutions. [Wikipedia] Three primary acids are found in wine grapes: RESVERATROL DERIVATIVES trans cis Malvidin-3-glucoeide Procyanidin B1 Quercetin R = H; resueratrol R = glucose; p Ice Id TYPICAL WINE FLAVONOIDS Resveratrol (3,5,4'-trihydroxy- trans-stilbene) is a stilbenoid, a type of natural phenol, and a phytoalexin produced naturally by several plants. [Wikipedia] In red wine, up to 90% of the wine's phenolic content falls under the classification of flavonoids. These phenols, mainly derived from the stems, seeds and skins are often leached out of the grape during the maceration period of winemaking. These compounds contribute to the astringency, color and mouthfeel of the wine. [Wikipedia] Prolog 2
  3. 3. Contents PART IV – QUANTUM FIELDS Review of Quantum Mechanics Galilean Invariance Lorentz Invariance The Relativity Principle Poincaré Transformations The Poincaré Algebra Lorentz Transformations Lorentz Invariant Scalar Klein-Gordon & Dirac One-Particle States Wigner’s Little Group Normalization Factor Mass Positive-Definite Boosts & Rotations Mass Zero The Klein-Gordon Equation The Dirac Equation References “It is more important to have beauty in one’s equations than to have them fit experiment … because the discrepency may be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory….” Paul Dirac, Scientific American, May 1963. 2017 MRT Determining the structure of the proton: a Feynman diagram for deep inelastic scattering process. The diagram shows the flow of momentum when a high energy electron e (••••) scatters (hence the exchange of a photon γγγγ with momentum q) from a quark (••••) taken from the wavefunction of the proton p (••••). This is a simple case called the Parton Model invented by Richard Feynman. We assume that the parton ( ) has negligeable (i.e., a small fraction ξ of ) transverse momentum with respect to the proton p, so the parton momentum ξ p is in the same direction as the proton momentum p, that is, the parton has momentum ξ pµ , where 0≤ξ ≤1. Finally, momentum conservation forces us to have the equality p′=ξ pµ + q given vertex couplings of the form ±ieγ µ where the gamma matrices satisfy γ µ γ ν + γ νγ µ = 2g µν. 3 Field (i.e., an interaction subjected to a potential Aµ
  4. 4. 2017 MRT PART V – THE HYDROGEN ATOM What happens at 10−−−−10 m? The Hydrogen Atom Spin-Orbit Coupling Other Interactions Magnetic & Electric Fields Hyperfine Interactions Multi-Electron Atoms and Molecules Appendix - Interactions The Harmonic Oscillator Electromagnetic Interactions Quantization of the Radiation Field Transition Probabilities Einstein’s Coefficients Planck’s Law A Note on Line Broadening The Photoelectric Effect Higher Order Electromagnetic Interactions References “Quantum field theory is the way it is because […] this is the only way to reconcile the principles of quantum mechanics […] with those of special relativity. […] The reason that quantum field theory describes physics at accessible energies is that any relativistic quantum theory will look at sufficiently low energy like a quantum field theory.” Steven Weinberg, Preface to The Quantum Theory of Fields, Vol. I. PART III – QUANTUM MECHANICS Introduction Symmetries and Probabilities Angular Momentum Quantum Behavior Postulates Quantum Angular Momentum Spherical Harmonics Spin Angular Momentum Total Angular Momentum Momentum Coupling General Propagator Free Particle Propagator Wave Packets Non-Relativistic Particle Appendix: Why Quantum? References 4
  5. 5. Review of Quantum Mechanics We will provide only the briefest of summaries of PART III–QUANTUM MECHANICS, in the generalized version of Dirac. This will also strengthen our mathematical conventions. It has a norm; for any pair of vectors there is a complex number 〈ΦΦΦΦ|ΨΨΨΨ〉, such that: ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦΦΦΦΦ ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦΨΨΨΨΨΨΨΨΦΦΦΦ ΦΦΦΦΨΨΨΨΨΨΨΨΦΦΦΦ 2 * 21 * 12211 22112211 * ηηηη ξξξξ +=+ +=+ = POSTULATE #1: Physical states are represented by rays in Hilbert space. where the asterisk (∗) indicates that the complex conjugate is taken. The norm 〈ΨΨΨΨ|ΨΨΨΨ〉 also satisfies a positivity condition: A ray is a set of normalized vectors: A Hilbert space is a kind of complex vector space; that is, if |ΦΦΦΦ〉 and |ΨΨΨΨ〉 are vectors in the space (often called ‘state-vectors’ or ‘kets’) then so is η|ΦΦΦΦ〉+ξ|ΨΨΨΨ〉, for arbitrary complex numbers η and ξ. 0≥ΨΨΨΨΨΨΨΨ and vanishes if and only if the state-vector (or ket) is null: |ΨΨΨΨ〉=0. with |ΨΨΨΨ〉 and |ΨΨΨΨ〉 belonging to the same ray if |ΨΨΨΨ〉=ξ|ΨΨΨΨ〉, where ξ is (as above) an arbitrary complex number with the extra condition that it’s magnitude is unity: |ξ |=1. 1=ΨΨΨΨΨΨΨΨ 2017 MRT 5
  6. 6. POSTULATE #2: Observables are represented by Hermitian operators. The above relation satisfies the reality condition: ΦΦΦΦΨΨΨΨΦΦΦΦΨΨΨΨ AAA ηξηξ +=+ These are mappings |ΨΨΨΨ〉→A|ΨΨΨΨ〉 of Hilbert space into itself, linear in the sense that: *† ΦΦΦΦΨΨΨΨΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦΦΦΦΦΦΦΦΦ AAA ===′ AA =† ( )Rinfor ΨΨΨΨΨΨΨΨΨΨΨΨ α=A The state represented by a ray R has a definite value α for the observable represented by an operator A if vectors |ΨΨΨΨ〉 belonging to this ray are eigenvalues of A with eigenvalue α: An elementary theorem tells us that for A Hermitian, α is real, and eigenvalues with different αs are orthogonal. 2017 MRT where the daggar (†) definies the Hermitian Operator and it indicates that the complex conjugate (i.e., replacing i → −i) and transpose (i.e., matrix elements are ‘transposed’ on either side of the diagonal, Aij →Aji ), A†=A*T, is taken. For any linear operator A the adjoint A† is defined by the scalar product of |ΦΦΦΦ〉 and |ΦΦΦΦ′〉=|A†ΨΨΨΨ〉: 6
  7. 7. POSTULATE #3: If a system is in a state represented by a ray R, and an experiment is done to test whether it is in any one of the different states represented by mutually orthogonal rays R1, R2, …, then the probability of finding it in the state represented by Rn is: where |ΨΨΨΨ〉 and |ΨΨΨΨn〉 are any vectors belonging to rays R and Rn, respectively. 2 )( nnP ΨΨΨΨΨΨΨΨ=→ RR Another elementary theorem gives a total probability unity (i.e., they add up to 100%): if the state-vectors |ΨΨΨΨn〉 form a complete set. 1)( =→∑n nP RR 2017 MRT 7
  8. 8. Let us now review symmetries: A symmetry transformation is a change in our point of view that does not change the result of possible experiments. For any such transformation R → R of rays we may define an operator U on Hilbert space H , such that if |ΨΨΨΨ〉 is in ray R then U|ΨΨΨΨ〉 is in the ray R, with U either unitary and linear: or else antiunitary and antilinear: If an inertial observer O (i.e., he ain’t movin’) sees a system in a state represented by a ray R or R1 or R2, …, then an equivalent observer O (i.e., he’s moving away!) who looks at the same system will observe it in a different state, represented by R or R1 or R2, …, respectively, but the two observers must find the same probabilities: )()( nOnO PP RRRR →=→ ΦΦΦΦΨΨΨΨΦΦΦΦΨΨΨΨ ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ UUU UU βαβα +=+ = ΦΦΦΦΨΨΨΨΦΦΦΦΨΨΨΨ ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ UUU UU ** * βαβα +=+ = 2017 MRT This is a fundamental theorem from Wigner (1931): Any symmetry transformation can be represented on the Hilbert spaceof physical states by an operator that is either linear and unitary or antilinear and antiunitary. 8
  9. 9. As mentionned in the condition 〈ΦΦΦΦ|A†ΨΨΨΨ〉=〈AΦΦΦΦ|ΨΨΨΨ〉=〈ΨΨΨΨ|A†ΦΦΦΦ〉∗ above, the adjoint of a linear operator L is defined by: With this definition, the conditions for unitarity or antiunitarity both take the form: ΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ LL ≡† ΦΦΦΦΨΨΨΨΨΨΨΨΦΦΦΦΨΨΨΨΦΦΦΦ AAA =≡ *† 1=UU† and there is always a trivial symmetry transformation R → R, represented by the identity operator U=1. This operator is, of course, unitary and linear. Continuity then demands that any symmetry (e.g., a rotation or translation or Lorentz transformation) that can be made trivial by a continuous change of some parameters (i.e., like angles or distances or velocities) must be represented by a linear unitary operator U (rather than one that is antilinear and antiunitary). This condition cannot be satisfied for an antilinear operator, because in this case the right-hand side of the above equation would be linear in |ΦΦΦΦ〉, while the left-hand side is antilinear in |ΦΦΦΦ〉. Instead, the adjoint of an antilinear operator A is defined by: 2017 MRT 1† − =UU Multiplying this unitary operator by it’s inverse U−1 wegetU†(UU−1)=1⊗U−1 =U−1. Thus: 9
  10. 10. A symmetry transformation that is infinitesimally close to being trivial can be re- presented by a linear ‘unitary operator’ that is infinitesimally close to the identity: The set of symmetry transformations has certain properties that define it as a group. If T1 is a transformation that takes rays Rn into Rn, and T2 is another transformation that takes Rn into Rn, then the result of performing both transformations is another symmetry transformation, which we write T2T1 (T1 then T2) that takes Rn into Rn. Also, a symmetry transformation T which takes rays Rn into Rn has an inverse, written T−1, which takes Rn into Rn, and there is an identity transformation T =1, which leaves rays unchanged. with ε a real infinitesimal (e.g., an infinitesimal change in the coordinates dxµ or an angle dϕ). For this to be unitary and trivial, T must be Hermitian and linear, so it is a can- didate for an observable. Indeed, most (and perhaps all) of the observables of physics (e.g.,angularmomentumormomentum) arise in this way from symmetry transformations. For φ =0, U(T) furnishes a representation of the group of symmetry transformations. 2017 MRT The unitary operators U(T) act on vectors in the Hilbert space, rather than on rays. If T1 takes Rn into Rn, then acting on a vector |ΨΨΨΨn〉 in the ray Rn, U(T1) must yield a vector U(T1)|ΨΨΨΨn 〉 in the ray Rn, and if T2 takes this ray into Rn, then acting on U(T1)|ΨΨΨΨn〉 it must yield a vector U(T2)U(T1)|ΨΨΨΨn〉 (again U(T1) then U(T2) but this time on |ΨΨΨΨn〉) in the ray Rn. But U(T2T1)|ΨΨΨΨn〉 is also in this ray, so these vectors can differ only by a phase φn(T2,T1) : )(e)()()(e)()( 12 ),( 1212 ),( 12 1212 TTUTUTUTTUTUTU TTi n TTi n nn φφ =⇒= ΨΨΨΨΨΨΨΨ )( 2 εε OTiU ++= 1 10
  11. 11. where Λµ ν is a constant matrix (a function of the velocity v of a ‘moving’ frame). Under an infinitesimal transformation of the variable θ, the coordinate differential dxµ is given by:         ==≡+Λ=→ ∑= c xfaxxx v v βζµµ ν ν ν µµµ tanh);()]([ 3 0 θθθθ θ θ θ µ dxd 0= ∂ ∂ ≡ all );( θθθθµµµµ xf so that the state vector |ψ 〉 will transform according to (i.e., by using Taylor’s expansion): As an example of symmetry,consideratransformation (parametrizedbythevariableθ, e.g., an angle ϕ,a translation a or aLorentzboostζ )onthespace-timecoordinates xµ : in which the real infinitesimal is ε =dθ and the generator for the parameter θ is given by: ∑= = ∂ ∂ ∂ ∂ −= 3 0 0 );( )( µ µ θ µ θ θ θ x xf iT all ψεεψθψ θ θ θψ ψ θ θ θψψψψψψ µµ µ µ θ µ µ µ µ µ θ µ µµ µ µ µµµµµ )]([)()()( );( )( )( );( )()()()()( 2 0 0 OTixTdix x xf idix x x xf dxx x dxxxdxx T ++=+=         ∂ ∂ ∂ ∂ −+= ∂ ∂ ∂ ∂ +≡ ∂ ∂ +≈+=→ ∑ ∑∑ = = 111 44444 344444 21 )(Generator all all 2017 MRT 11
  12. 12. with f a(θ ,θ) a function of the θs and θs. Taking θ a =0 as the coordinate of the identity, we must have: )),(()()( θθθθ fTTT = aaa ff θθθ == ),0()0,( As mentionned above, the transformation of such continuous groups must be represen- ted on the physical Hilbert space by unitary operators U[T(θ )]. For a Lie group these op- erators can be represented by a power series (e.g., in the neighborhoodof the identity): A finite set of real continuous parameters θ a describe a group of transformations T(θ ) with each element of the group connected to the identity by a path (i.e., UU−1=1) within the group. The group multiplication law U(T2)U(T1)=U(T2T1) thus takes on the form (i.e., a connected Lie group): According to f a(θ ,0)= f a(0 ,θ)=θ a above, the expansion of f a(θ ,θ) to second-order must take the form: 2017 MRT K+++= ∑∑∑ c b bc cb a a a TTiTU θθθθ 2 1 )]([ 1 where Ta, Tbc =Tcb, &c. are Hermitian operators independent of the θ s. Suppose that the U[T(θ )] form an ordinary (i.e., φ =0) representation of this group of transformations, i.e.: ))],(([)]([)]([ θθθθ fTUTUTU = ...),( +⊕+= ∑∑c b cba bc aaa ff θθθθθθ with real coefficients f a bc. The addition of the second-order term is emphasized by ⊕. 12
  13. 13. (The Σ were also omitted to get space.) The terms of order 1, θ, θ, and θ 2 automatically match on both sides of this equation – from the θ θ terms we get a non-trivial condition: KKKK ++++++++=+++×+++ bc ccbb a cba bc aa bc cb a a bc cb a a TTfiTTitti ))(()(][][ 2 1 2 1 2 1 θθθθθθθθθθθθθθ 111 ∑−−= a a a bccbbc TfiTTT Since we are following Weinberg’s development, he points out that: This shows that if we are given the structure of the group, i.e., the function f a(θ,θ ), and hence its quadratic coefficient f a bc, we can calculate the second-order terms (i.e., Tbc) in U[T(θ)] from the generators Ta appearing in the first-order terms. (A pretty amazing fact wouldn’t you say?) Applying the multiplication rule U[T(θ)]U[T(θ)]=U[T( f (θ ,θ )] and using the series U[T(θ)]=1+iθ a Ta +½θ bθ c Tbc +…above withθ → f a (θ,θ)=θ a +θ a + f a bc θ b θ c ,we get: where Ca bc are a set of real constants known as structure constants: ∑=−≡ a a a bcbccbcb TCiTTTTTT ],[ 2017 MRT However, as he points out: There is a consistency condition: the operator Tbc, must be symmetricinbandc(becauseitisthesecondderivativeof U[T(θ)]withrespect toθ b andθ c) so the equation Tbc =−Tb Tc −iΣa f a bc Ta above requires that the commutation relations be: a cb a bc a bc ffC +−= Such a set of commutation relations is known as a Lie algebra to mathematicians. 13
  14. 14. This is the case for instance for ‘translations’ in spacetime, or for ‘rotations’ about any one fixed axis(though not both together).Then the coefficients f a bc in the function f a(θ,θ ) =θ a +θ a +Σbc f a bc θ b θ c vanish,and so do the structure constants Ca bc=− f a bc+ f a cb, that is Ca bc =0. So, [Tb ,Tc]≡TbTc −TcTb reduces to the fact that the generators then all commute: aaa f θθθθ +=),( N N TUTU                   = θ θ)]([ Such a group is called Abelian. In this case, it is easy to calculate U[T(θ)] for all θ a. Again, from the group multiplication rule U[T(θ)]U[T(θ)]=U[T( f (θ ,θ ))] and the function f a(θ,θ)= θ a +θ a above, and taking ε=θ/N, we have for any integer N: As a special case of importance, suppose that the function f a(θ,θ) is simply additive: and hence: 0],[ =cb TT 2017 MRT Letting N→∞, and keeping only the first-order term in U[T(θ /N)], we have then: N a a a N T N iTU                 += ∑∞→ θ θ 1lim)]([ TiTi UTU a a a θθ θθ e)(e)]([ =⇒= ∑ 14
  15. 15. Lorentz invariance is needed to replace the principle of Galilean invariance and the discovery of the non-conservation of parity in weak interactions (1956) has reemphasized that an invariance principle and its consequences must be experimentally verified. One key invariance principle in quantum mechanics and quantum field theory is that (c.f., Review of Quantum Mechanics chapter): Different equivalent observers make the same predictions as to the outcome of an experiment carried out on a system. The vectors |φO〉 and |ψO〉 seen by observer O and vectors |φO〉 and |ψO〉 seen by observer O. A unitary transformation U(L) (a function of the Lorentz transformation) relates both systems. |φO〉 |ψO〉 We shall call the vector |ψO 〉 the translation of the vector |ψO〉. Stated mathematically, the postulate above asserts that if |ψO〉 and |φO 〉 are two states and |ψO 〉 and |φO 〉 their translations, then: OO LU ψψ )(= where U depends on the coordinate systems between which it affects the correspondence and U(L ≡1)=1 if L is the identity transformation 1 (i.e., if O and O are the same coordinate system). If all rays in Hilbert space are distinguishable, it the follows from the above equation – as a mathematical theorem (Wigner, 1931) – that the correspondence |ψO 〉 → |ψO 〉 is effected by a unitary or anti-unitary operator, U(O,O), the operator U is completely determined up to a factor of modulus 1 by the transformation L which carries O in O. We write: 22 OOOO ψφψφ = |φO〉 |ψO〉 O O U(L) OOOO OO LU LU ψφψφ ψψ )( )( = = 2017 MRT This statement means that observer O will attribute the vector |ψO〉 to the state of the system, whereas observer O will describe the state of this same system by a vector |ψO〉. 〈φO |U(L)|ψO〉 Lorentz Invariance 15
  16. 16. U(v) |ΨΨΨΨO 〉 |ΨΨΨΨO 〉p σ v OO z,z sees sees y x p m x For special relativity, as an example, we consider the inhomogeneous Lorentz transformations. A relativity invariance requires the vector space describing the possible states of a quantum mechanical system to be invariant under all relativity transformations (i.e., it must contain together with every |ψ 〉 all transformations U(L)|ψ 〉 where L is any special relativity transformation). The transformed states can always be obtained from the original state by an actual physical operation on the system. Consider for example a Lorentz transformation along the z-axis with velocity v. The transformed state, which arises from the momentum eigenstate |ΨΨΨΨO(p,σ )〉, is given by U(v)|ΨΨΨΨO(p,σ )〉. This is the state of the system as seen by observer O. It is, however, also a possible state of the system as seen by O and which can be realized by giving the system a velocity −v along the z-axis. The state vector |ΨΨΨΨO〉 seen by observer O and the vector | ΨΨΨΨO 〉 seen by observer O moving away from O at velocity v. A unitary trans- formation U(v) brings state |ΨΨΨΨO〉 into state |ΨΨΨΨO 〉. Here are two typical problems*: 1. Suppose that observer O sees a W-boson (spin one and mass m≠0) with momentum p in the y-direction and spin z-component σ . A second observer O moves relative to the first with velocity v in the z-direction. How does O describe the W state? 2. Suppose that observer O sees a photon with momentum p in the y-direction and polarization vector in the z-direction. A second observer O moves relative to the first with velocity v in the z-direction. How does O describe the same photon? 2017 MRT * S. Weinberg, Quantum Theory of Fields, Vol. I – Foundations, 1995 – P. 104-105. Solving these two problems is the goal of the following slides and they involve modern concepts starting with particle definitions as unitary representations. 16
  17. 17. The Relativity Principle Einstein’s principle of relativity states the equivalence of certain ‘inertial’ frames of reference. It is distinguished from the Galilean principle of relativity, obeyed by Newtonian mechanics, by the transformation connecting coordinate systems in different inertial frames. 2017 MRT Here ηµν is the diagonal 4×4 matrix, with elements (i.e., the Minkowski metric): η00 =+1, η11 =η22 =η33 =−1 and ηµν ≡0 for µ ≠ν. These transformations have the special property that the speed of light c is the same in all inertial frames (e.g., a light wave traveling at speed c satisfies |dr/dt|=c or in other words Σµνηµν dxµ dxν =c2dt2 − dr2 =0, from which it follows that Σµνηµν dxµ dxν =0, and hence |dr /dt|=c). If the contravariant vector xµ =(ct,r) are the coordinates in one inertial frame (x1, x2, x3) [i.e., r = xi (i=1,2,3), as Cartesian space coordinates, and x0 =ct a time coordinate, the speed of light being c] then in any other inertial frame, the coordinates xµ must satisfy: OO xdxdxdxd FrameFrame ∑∑∑∑ = == = ≡ 3 0 3 0 3 0 3 0 µ ν νµ µν µ ν νµ µν ηη or equivalently stated as the Principle of covariance: ∑∑ ∂ ∂ ∂ ∂ = µ ν µνσ µ ρ µ σρ ηη x x x x The covariant vector can be given as xµ=Σνηµν xν =(ct,−r). The norm of the vector Σµ xµ xµ =(x0)−Σi(xi)2 =c2t2 −|r|2 is a Lorentz invariant term. 17
  18. 18. Poincaré Transformations with aµ arbitraryconstants(e.g.,‘leaps’),and Λµ ν aconstantmatrixsatisfyingthecondition: Any coordinate transformation xµ → xµ that satisfies Σµνηµν (∂xµ /∂xρ )(∂xν /∂xσ )=ηρσ is linear and allows us to define the Poincaré Transformations: 2017 MRT µµµµµµ ν ν ν µµ axxxxaxx +Λ+Λ+Λ+Λ=+Λ= ∑= 3 3 2 2 1 1 0 0 3 0 σρ µ ν σ ν ρ µ µν ηη =ΛΛ∑∑ The matrix ηµν has an inverse, written ηµν , which happens to have the same components: it is also diagonal matrix, with elements: η00 =+1,η11 =η22 =η33 =−1 and ηµν ≡0 for µ ≠ν. To save on using summation signs (Σ), we introduce the Summation Convention: We sum over any space-time index like µ and ν (or i, j or k in three dimensions) which appears twice in the same term, if they appear only once ‘up’ and also only once ‘down’. As an example, and while also enforcing this tricky summation convention, we now multiply ηµµµµνννν Λµµµµ ρ Λνννν σ=ηρσ above with ησττττΛκ ττττ and when inserting parentheses for show: ρ κ ρ κκ ρ κ ρ κ ρ κ ρ ηηηηηηηηη µµµµνννν µνµνµνµν σσσσ σσσσ ττττσσσσ ττττσσσσ ττττσσσσ ττττσσσσ ννννµµµµ µνµνµνµν µνµνµνµν ττττσσσσ ττττσσσσ ττττσσσσ ννννµµµµ µνµνµνµν Λ=Λ=Λ=Λ=ΛΛΛ=ΛΛΛ∑ ∑ ][)(])[(])[( 44 34421 and now, when multiplying with the inverse of the matrix ηµµµµν Λµµµµ ρ, we get from this: ττττσσσσ ττττσσσσ ηη κνκν ΛΛ= ( )3,2,1,0=µ 18
  19. 19. 2017 MRT ( )νρaxaxx andoversum,ρρρρνννν νννν ρρρρ ρρρρ ρρρρ ρρρρ i.e.)( +ΛΛ=+Λ= µµµµ These transformations do form a group. If we first perform a Poincaré transformation xµ → Λµ ν xν +aµ, and then a second Poincaré transformation xµ → xµ, with: )()( µρ ρ µν ν ρ ρ µµ aaxx +Λ+ΛΛ= then the effect is the same as the Poincaré transformation xµ → xµ, with: Taking the determinant of ηµν Λµ ρ Λν σ=ηρσ gives: so Λµ ν has an inverse, (Λ−1)ν σ, which we see from ηµν Λµ ρ Λν σ=ηρσ and takes the form: { σ µσρ µν ρ νν ρ ηη Λ=Λ=Λ− N.B. )( 1 1)Det( 2 =Λ The transformation T =T(Λ,a) induced on physical states therefore satisfy the group composition rule: ),(),(),( aaTaTaT +ΛΛΛ=ΛΛ and the inverse of this T(Λ,a) transformation is also obtained from T(Λ,a)T(Λ,a)= T(ΛΛ,Λa+a) above to be T(Λ−1,−Λ−1a) such that: The whole group of transformations T(Λ,a) is properly known as the inhomogeneous Lorentz group, or Poincaré group. It has a number of important subgroups – notably T(Λ,0) which we will look at in greater detail in a little while. 1=Λ−ΛΛ −− ),(),( 11 aTaT 19
  20. 20. and this unitary operator U satisfies the same group composition rule as T(Λ,a): So, in accordance with the discussion in the previous slides, the transformations T(Λ,a) induced a unitary linear transformation on a state vector in the Hilbert space: For example, we will soon discuss the wave function in its momentum representation: for which the same composition rule applies: 2017 MRT )]()([)]([)]([),( 1 pLpLUpLUpLUpU ΛΛΛ=Λ≡Λ − ),(),(),(),;( jjj mppUmpmjp ΨΨΨΨΨΨΨΨΨΨΨΨ Λ=→µ )(),()()( xaUxx ψψψ µ Λ=→ ),(),(),( aaUaUaU +ΛΛΛ=ΛΛ 20
  21. 21. Now it is time to study three-dimensional rotations and add relativity to the overall description. To this effect we will exploit pretty much all the group symmetry properties! 2017 MRT ζζβγ ζζβγ sinhcosh)( sinhcosh)( 03033 22 11 30300 xxxxx xx xx xxxxx −=−= = = −=−= and v=|v| is the relative velocity of the two frames. or (N.B., implicit sum on ν ): β ζ ζ ζβγβζ β ζγ ==      == − == cosh sinh tanhsinh 1 1 cosh 2 aswellaswithand c v                           − − =               3 2 1 0 3 2 1 0 cosh00sinh 0100 0010 sinh00cosh x x x x x x x x ζζ ζζ that is, assuming propagation in the direction of the x3-axis. This can be represented in matrix form as: Let us recall a few facts about homogeneous Lorentz transformations: where: ν ν µνµµ ζ xxxx )],-([Λ= 21
  22. 22. The explicit matrix representation of a restricted homogeneous Lorentz transformation in the x1-direction (i.e., a rotation in the x0-x1 plane) is given by: Similarly, the infinitesimal generators M02 and M03 for rotations in the x0-x2 and x0-x3 planes respectively, are given by: and the infinitesimal generator M10 for this rotation is defined as:             − − = 1000 0100 00coshsinh 00sinhcosh )(Λ[01] ζζ ζζ ζ             ==≡ = 0000 0000 0001 0010 )(Λ 0 [01] 10 1 ζ ζ ζ d d MK             ==≡             ==≡ == 0001 0000 0000 1000 )(Λ 0000 0001 0000 0100 )(Λ 0 [03] 03 3 0 [02] 02 2 ζζ ζ ζ ζ ζ d d MK d d MK , 2017 MRT 22
  23. 23. The infinitesimal generators in the xi-xj plane, i.e. spatial rotations, are: 2017 MRT             − =≡             − =≡             − =≡ 0000 0010 0100 0000 0010 0000 1000 0000 0100 1000 0000 0000 12 3 31 2 23 1 MJMJMJ and, where we define Mµν =−Mνµ. In matrix form these terms come together as:             −− − =               −− − = 0 0 0 0 0 0 0 0 123 132 231 321 231303 231202 131201 030201 JJK JJK JJK KKK MMM MMM MMM MMM M µν 23
  24. 24. The general result for Mµν can now be written alternatively as: 2017 MRT where we can use ηµν Λµ ρ Λν σ=ηρσ to show that ωµν (or ωµ ν ) is antisymmetric ωµν =−ωνµ : µν µν M i ω 2 1 1)ω(Λ h += which implies ωρσ +ωσρ =0 QED. An arbitrary infinitesimal Lorentz transformation, by expanding according to Λ(ω)= exp(−½iMµν ωµν /h) in a power series, can be written as: KζSω •−•=L and KζSω •−• = eA where ωωωω and ζζζζ are constant 3-vectors. ρσσρσρ ρ µ σµσ ν νρσρ σ ν ρ µ σ ν ρ µ σ ν ρ µ µν σ ν σ ν ρ µ ρ µ µνσρ η ηηη δδδδη δδηη ωω ωω )ωω( )ω)(ω( ++= ++= ++= ++= 24
  25. 25. A finite rotation in the µ -ν plane (in the sense µ toν ), is again obtained by exponentiation: 2017 MRT ζνµ µν ζ M xx e),-(Λ = µσνρνρµσµρνσνσµρρσµν ηηηη MMMMMM −−+=],[ One verifies that the infinitesimal generators, Mµν ,satisfy the following commutation rules: 25
  26. 26.         ∂ ∂ = ∂ ∂ −= ∂ ∂ −= ∂ ∂ ∂ ∂ −=• = kj kji ikji kji k k k x xi x xi x i x xf i εε ζ ζ ϕ µ )ˆ()ˆ()ˆ( );(ˆˆ 0 nnrn Jn ×××× hence: kj kjii x xiJ ∂ ∂ = ε For the rotation acting on the space-time coordinates, note that the time coordinate is unaffected (hence only latin indices):    •−= −•−+= = +−+==≡ ζζ ζζ ϕϕϕϕµµ sinhˆcosh ]sinhˆ)1[(coshˆ ]ˆ[sin)]ˆ(ˆ)[cos1()],ˆ([);( rn nrnrr rnrnnn cttc ct xxRxfx kkki i k ×××××××××××× in which we made use of the relation xj =ηij xj =−x j, η being the Minkowski 3-metric. It can be shown that the generators for rotation are equivalent to the generators for the SO(3) Special Orthogonal group (which are Hermitian). Thus, the representation for a finite rotation acting on the wavefunction is unitary and it is given by U(R)=exp(−iϕ n•J/h).ˆ 2017 MRT where v=ctanhζ n. So, we get:ˆ 26
  27. 27. The general Lorentz transformations for a simple spatial rotation, ΛR, is given by: in which the three-dimensional spatial rotations, Rn(ϕ), are elements in the simple orthogonal group SO(3). However, this is not relavant for evaluating the Wigner coefficients since it is trivial to show that both the Wigner transformation and Lorentz rotation, ΛR, belong to the same little group H(Λ,k). µν µν ω 2 1 e)ω( M i h − =Λ Note that the parameters associated with the Lorentz transformation are given by the anti-symmetric tensor ωµν. (In addition, the matrix representations of the Lorentz generators in the four-vector coordinates are given by:       − =      = iab i ai bi i i J i i K εδ δ 0 00 0 0 and where the indices a, b represent the rows and columns, respectively.) The convenional way of characterizing the Lorentz transformation, Λ, is described by the generators for boosts with Ki =M0i =−Mi0 and rotations Ji =½ε ijk Mjk, that is: ˆ             =Λ 0 )]([0 0 0001 )( ˆ j iR R ϕ ν µ n 2017 MRT 27
  28. 28. in which we recall that coshζ =γ and sinhζ =βγ where γ =1/√(1−β2) and β =v/c. In four- vector notation we have:               + + =                           =               ζζ ζζ ζζ ζζ coshsinh sinhcosh cosh00sinh 0100 0010 sinh00cosh 30 2 1 30 3 2 1 0 3 2 1 0 xx x x xx x x x x x x x x );()]([ ζζ µν ν µµ xfxx ≡Λ= and the associated generator is given as: Now, consider a Lorentz boost along the x3-axis: since Λ0 0|ζ =0 =coshζ |ζ =0=1, &c.       ∂ ∂ − ∂ ∂ =      ∂ ∂ + ∂ ∂ −=         ∂ ∂ ∂ Λ∂ + ∂ ∂ ∂ Λ∂ + ∂ ∂ ∂ Λ∂ + ∂ ∂ ∂ Λ∂ −= ∂ ∂ ∂ Λ∂ −= ∂ ∂ ∂ ∂ −= ==== == 30033 0 0 3 3 3 0 3 3 3 0 0 0 3 0 3 0 3 0 0 0 0 0 0 00 3 );( x x x xi x x x xi x x x x x x x xi x xi x xf iK ζζζζ µ ν ζ ν µ µ ζ µ ζζζζ ζζ ζ 2017 MRT 28
  29. 29. Similarly: )( 0000 iiiii xxi x x x xiK ∂−∂=      ∂ ∂ − ∂ ∂ = † 00 ])([)()()( xKx x xx x xixK iiii ψψψψ ≠      ∂ ∂ − ∂ ∂ = Consider that the state-vector of the system is given by |ψ 〉, hence the action of Ki on the state is: This implies that the generator for the Lorentz boost, Ki, is not Hermitian and hence the exponentiation of the generator (i.e., exp(−iζ i Ki /h)) will not be unitary. The representation of the Lorentz boost acting on the wavefunction is not unitary and hence is not trace-preserving. We can summarize the effects of the rotations and the Lorentz boost into one second- rank covariant tensor: )( µννµµν ∂−∂= xxiM in which Ji =½ε ijk Mjk and Ki =Mi0. 2017 MRT 29
  30. 30. These generators (i.e., Pµ and Mµν ) obey the following commutation relations, which characterize the Lie algebra of the Poincaré group (and adding h and c for reference): kkjijikkjijikkjiji JiKKKiKJJiJJ εεε hhh −=== ],[],[],[ and; 0],[ )(],[ )(],[ = −−= −+−−= λµ νµλµνλλµν νρµσµρνσµσνρνσµρµνρσ ηη ηηηη PP PPiPM MMMMiMM h h The rotation Ji and boost Kj generators can be written in covariant notation Mµν and the commutation relations are then re-written as: 2017 MRT 00 ],[],[],[ P c iPKcPiPKPiPJ jijiiikkjiji δε h hh === and; 0],[0],[0],[ 00 === jiii PPPPPJ and; The first of these is the usual set of commutators for angular momentum, the second says that the boost K transforms as a three-vector under rotations, and the third implies that a series of boosts can be equivalent to a rotation. Next we have: where P0c=H, the Hamiltonian, and finally all components of Pµ should commute with each other: Together, these equations above form the Lie algebra of the Poincaré group. 30
  31. 31. we get: µµµµµ axaxfx +=≡ );(    ≠= == = ∂ ∂ = νµ νµ δ ν µ ν µ µ when when 0 1);( 0a a axf and using: ννµν µ ν δ ∂−= ∂ ∂ −= ∂ ∂ −= hhh i x i x iP For a simple translation aµ of the coordinates xµ: that is, the momentum operator. For a finite translation aµ, the state-vector |ψ 〉 of a relativistic system (expressed by the Dirac wavefunctionψ ) will transform as: )(e)()( xxx aP i ψψψ µ µ µ h − =→ Note that the generators P0 =−ih∂0 and Pk = −ih∂k are known as the Hamiltonian and momentum operators and they are Hermitian since their associated eigenvalues are defined to be real. This implies that the representation for a finite translation acting on the wavefunction (e.g., as you will see soon it is given by U(1,a)=exp(−iPµ aµ /h)) is actually unitary. 2017 MRT ψεψεψψ µν µ ν       +=         ∂ ∂ ∂ ∂ −+= = P i xa axf i i a h h h 11 0 );( all we obtain: 31
  32. 32. µ µ µ aP i aIU h − × = e),( 44 00 ωωω iiiji kjik −=== ζεϕ and 0 2 1 iikj kjii MKMJ == andε The contravariant generators for the space-time translations aµ are defined by Pµ, in which the time translation, P0 and the spatial translation, Pi are the Hamiltonian and momentum operators, respectively, of a free particle. The finite translations acting on the space-time coordinates are well-defined (with no Lorentz transformation): µν µν ω 2 1 e)ω( M i h − =Λ The Lorentz transformation can be described by its generators (without translation): Here ζ i is the Lorentz boost along the i-th axis, and θk are the parameters involved in the rotation along the axial vector. The generators for these Lorentz transformations are given by Mµν have been explicitely derived previously. Here the generators for the spatial rotation Ji and Lorentz boost Ki are given by: with the corresponding covariant generators Pµ =ηµν Pν . The Lorentz transformation Λ can be described by an antisymmetric second-rank tensor ωµν which is defined by the parameters in Λ: 2017 MRT 32
  33. 33. and it produces the following space-time coordinate transformation: In general, the elements of the Poincaré group are given as:       +−−− ==Λ µν µν µ µ µ µ µν µν ω 2 1 ω 2 1 eee),( MaP i aP i M i aU hhh in which Pµ is the momentum of the particle in the new coordinate frame. µν ν µµµ axxx +Λ=→ µν ν µµµ PPPP +Λ=′→ Similarly, the momentum of a free particle also transforms according to: By definition, a momentum contravariant four-vector is given by Pµ Pµ =(P0/c)2 −(Pi )2 = (E/c)2−|p|2 =mo 2c2 is another Lorentz invariant term as well. In this context, mo is defined as the rest mass of the particle. 2017 MRT 33
  34. 34. For the inhomogeneous Lorentz group, the identityis the transformation Λµ ν=δ µ ν , aµ =0, so we want to study those transformations with: µµ ν µ ν µ ν µ εδ =+=Λ aandω where ωσρ≡ηµσ ωµ ρ and ωµ ρ≡ηµσ ωσρ. Keeping only the terms of first order in ω in the Lorents condition ηµν Λµ ρ Λν σ=ηρσ , we see that this condition now reduces to the antisymmetry of ωσρ : both ωµ ν and ε µ being taken as infinitesimal. The Lorentz condition ηµν Λµ ρ Λν σ=ηρσ reads here: An antisymmetric second-rank tensor in four dimensions has (4×3)/2=6 independent components, so including the four components of ε µ, an inhomogeneous Lorentz transformation is described by 6+4=10 parameters. 2017 MRT 34 ρσσρ ωω −= )ω(ωω )ω()ω( 2 O+++= ++= ΛΛ= σρρσρσ σ ν σ ν ρ µ ρ µ µν σ ν ρ µ µνσρ η δδη ηη
  35. 35. Since U(1,0) carries any ray into itself, it must be proportional to the unit operator, and by a choice of phase may be made equal to it.For an infinitesimal Lorentz transformation Λµ ν=δ µ ν +ωµ ν andaµ =εµ,U(1+ω,ε) must then equal 1 plus terms linear in ωρσ and ερ .We write this as: ...ω 2 1 ),ω( ++−=+ σρ σρ ρ ρεε M i P i U hh 11 σρσρρρ MMPP == †† )()( and Since ωρσ is antisymmetric, we can take its coefficients Mρσ to be antisymmetric also: Here Pρ and Mρσ are ε- and ω-independent operators, respectively, and the dots denote terms of higher order in ε and/or ω. In order for U(1+ω,ε) to be unitary, the operators Pρ and Mρσ must be Hermitian: 2017 MRT As we shall see, P1, P2, and P3 are the components of the momentum operator, M23, M31, and M12 are the components of the angular momentum vector, and P0 is the energy operator, or Hamiltonian. ρσσρ MM −= 35
  36. 36. ]ω,ω)([ ]ω)(,ω)([ ]ω)(,ω)([ ]ω)(,ω)[(),( ])ω)((,ω)[(),(),(),ω(),(),(),ω(),( 11 111 11 ω)(ω)( 11 11 ω 111 11 11 aU aaaU aaU aUaU aUaUaUUaUaUUaU aaaa aaa −− −−− −− +Λ+−=Λ+=Λ=Λ=Λ −− −− Λ−=Λ=Λ=+=Λ −−− ΛΛ−ΛΛ+Λ≡ ΛΛ+Λ+ΛΛ−Λ+Λ= +Λ+Λ+Λ−Λ+Λ= +Λ+−Λ+Λ= +Λ−+Λ+Λ=Λ−Λ+Λ=Λ+Λ −− −− ε ε ε ε εεε ε ε 1 1 11 11 1111 11 1 4444444 34444444 21 4444 34444 21 &;; &;; 2017 MRT where Λµ ν and aµ are here the parameters of a new transformation, unrelated to ω and ε. Let us consider the Lorentz transformation properties of Pρ and Mρσ. We consider the product: ),(),( 1 aUaU Λ+Λ − ),(1 εεεεωωωωU In the end the transformation rule is given by: ],[),(),(),( 111 aUaUUaU −−− ΛΛ−ΛΛΛ+≡Λ+Λ ωωωωωωωωωωωω εεεεεεεε 11 since Λ1Λ−1= 1. According to the composition rule T(Λ,a)T(Λ,a)= T(ΛΛ,Λa+a) with T =U(Λ,a) the product U(Λ−1,−Λ−1a) U(Λ,a) equals U(1,0), so U(Λ−1,−Λ−1a) = U−1(Λ,a), i.e., U(Λ−1,−Λ−1a) is the inverse of U(Λ,a). It follows from U(Λ,a)U(Λ,a)= U(ΛΛ,Λa+a) that, in sufficient detail to show these important group operations so that they be well understood, we have: 36
  37. 37. 2017 MRT Using U(1+ω,ε)=1−(i/h)ερ Pρ +½(i/h)ωρσ Mρσ to first order in ω and ε we have then: )(ω 2 1 )( 2 1 )( 2 1 ω ω 2 1 )(ω 2 1 ])(ω[ )ω( 2 1 )ω(),(ω 2 1 ),( 1 1 111 µννµµνσ ν ρ µρσ µρ µρ µννµνµµνσ ν ρ µρσ µρ µρ µνσ ν ρ µρσ µν ν σ ρσ ρ µ µν ν σ ρσ ρ µ µ ρ ρ µ µν µν µ µ σρ σρ ρ ρ ε ε ε εε PaPaM i P i PaPaPaPa i P i M i M i PaP i M i Pa i aUM i P i aU −−ΛΛ+Λ−=       ++−ΛΛ+ Λ− ΛΛ+= ΛΛ+ ΛΛ−Λ−= ΛΛ+ΛΛ−Λ−=Λ      +−Λ − − −−− hh h h h h h hhhh 1 1 1 11 Equating coefficients of ωρσ and ερ on both sides of this equation we find: )(),(),( ),(),( 1 1 µννµµνσ ν ρ µ σρ µρ µ ρ PaPaMaUMaU PaUPaU −−ΛΛ=ΛΛ Λ=ΛΛ − − where we have exploited the antisymmetry of ωρσ , i.e., ωρσ =−ωσρ , and we have used the inverse (Λ−1)ν σ =Λν ρ =ηνµ ηρσ Λµ ρ. 37
  38. 38. Next, let’s apply these rules rules to a transformation that is itself infinitesimal, i.e., Λµ ν=δ µ ν +ωµ ν and aµ =ε µ, with infinitesimals ωµ ν andεµ unrelated to the previous ω and ε. 2017 MRT Equating coefficients of ωρσ and ερ on both sides of these equations, we would find these commutation rules: µρνσρνµσσνρµµσρνρσµν ρσµσρµρσµ ρµ ηηηη ηη MMMMMM i PPMP i PP +−+= −= = ],[ ],[ 0],[ h h This is the Lie algebra of the Poincaré group. µρ µ ρµ µ µν µν ε PPPM i ω,ω 2 1 =      − h νρσ ν µσρ µ ρσσρµνµ µ µν µν εεε MMPPMPM i ωω,ω 2 1 +++−=      − h and By using U(1+ω,ε)=1−(i/h)ερ Pρ +½(i/h)ωρσ Mρσ and keeping only terms of first order in ωµ ν andε µ, our equations for Pρ and Mρσ become: 38
  39. 39. In quantum mechanics a special role is played by those operators that are conserved, i.e., that commute with the energy operator H=P0. We just saw that [Pµ,Pρ]=0 and (i/h)[Pµ,Mρσ ]=ηµρPσ−ηµσ Pρ shows that these are the momentum three-vectors: 2017 MRT and the angular momentum three-vector: These are not conserved, which is why we do not use the eigenvalues of K to label physical states. In a three-dimensional notation, the commutation relations may be written: where i, j, k, &c. run over the values 1, 2, and 3, and εijk is the totally antisymmetric quantity with ε123 =+1. The commutation relation [Ji ,Jj ]=iεijk Jk is the angular- momentum operator. ],,[ 321 PPP=P ],,[],,[ 211332123123 JJJJJJ −−−==J and the energy P0 itself. The remaining generators form what is called the ‘boost’ three- vector: ],,[],,[ 030201302010 JJJJJJ −−−==K jijikjkiji kjkijikjkijikjkiji iiii HiPKPiPJ JiKKKiKJJiJJ PiHKHHHPHJ δε εεε == −=== ==== ],[],[ ],[],[],[ ],[0],[],[],[ and and, and 39
  40. 40. Now,* there is one peculiar consequence to one of these commutators – the two boost generators are: 2017 MRT This commutator means that two boosts, Bi and Bj , in different directions (i.e., the indices i and j can’t equal each other at the same time) are not equivalent to a single boost B: where B is some boost. The reason things aren’t equal is the factor Wn××××m(Ω), the Wigner Rotation where Ω is the Wigner Angle (i.e., a true space-time rotation although to be realistic, for practical reason it is usually an infinitesimal one.) BWBB )(ˆˆˆˆ Ω= mnmn ×××× kjkiji JiKK ε−=],[ )()()()( ˆˆˆˆ 1 ˆˆˆˆˆˆ Ω=ΩΩΩ= − mnmnmnmnmn ×××××××××××××××× WBWBWWBB * Credit for developing this in the way it is shown here (and in the next few slides), with an example of which is given pretty much as it is, is much due to Entanglement in Relativistic Quantum Mechanics, E. Yakaboylu, arxiv:1005.0846v2, August 2010. By using B=WBW−1, the expression Bn Bm =Wn××××m (Ω)B above can be re-written as:ˆ ˆ ˆ ˆ ˆ ˆ 40
  41. 41. For example, let us use Lorentz transformations as boost matrices along the x- and y- direction (i.e., along x and y unit vectors), respectively, as defined by: 2017 MRT Notice that the result is non-symmetric. Now we can write this matrice for By Bx as: ff BWBWBB )()( ˆˆˆˆˆ Ω=Ω= −zxyxy ×××× where the ‘arbitrary’ Wigner Rotationmatrixhereis givenby(N.B., yes,aroundnegativez):             − − −− =             − −             − − = 1000 0 00 0 1000 0100 00 00 1000 00 0010 00 ˆˆˆˆˆˆˆˆ ˆˆˆ ˆˆˆˆˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆ yxyxyyxy xxx yyxxyxy xxx xxx yyy yyy xy γββγγβγγ γβγ βγβγγγγ γβγ βγγ γβγ βγγ BB         Ω=             ΩΩ− ΩΩ =Ω − − )(0 01 1000 0cossin0 0sincos0 0001 )( ˆ ˆ z z RW The orderedproduct (i.e.,reading“right-comes-after-left”as the product ByBx is applied)is:             − − =             − − = 1000 00 0010 00 1000 0100 00 00 ˆˆˆ ˆˆˆ ˆ ˆˆˆ ˆˆˆ ˆ yyy yyy y xxx xxx x γβγ βγγ γβγ βγγ BB and ˆ ˆ 41 ˆˆ ˆ ˆ ˆ
  42. 42. The result of the (group) matrix multiplication is (N.B., remember that M−1 =η MT η): 2017 MRT The Wigner angle Ω can be obtained by demanding that the Bf matrix be symmetric in, say, its M23 and M32 components: and after solving for the ratio sinΩ/cosΩ=tanΩ we get:             ΩΩ+ΩΩ−Ω− Ω−Ω−ΩΩ+Ω− −− =             − − −−             ΩΩ Ω−Ω =Ω= − − 1000 0coscossincossin 0sinsincossincos 0 1000 0 00 0 1000 0cossin0 0sincos0 0001 )( ˆˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆ ˆˆˆ ˆˆˆˆˆˆˆ ˆˆ 1 ˆ yxyxyxyxyxx yxyxyxyxyxx yyxxyxy yxyxyyxy xxx yyxxyxy xyz γββγγγβγγβγ γββγγγβγγβγ βγβγγγγ γββγγβγγ γβγ βγβγγγγ BBWBf Ω+Ω=Ω− cossinsin ˆˆˆˆˆˆ xyxyxy ββγγγγ xy xyxy ˆˆ ˆˆˆˆ tan γγ ββγγ + −=Ω or: 1 cos 1 sin ˆˆ ˆˆ ˆˆ ˆˆˆˆ + + −=Ω + −=Ω xy xy xy xyxy γγ γγ γγ ββγγ and 42
  43. 43. By replacing sinΩ and cosΩ in the boost matrix Bf one gets: 2017 MRT Notice now that this Bf matrix is symmetric. So, as a result in this case (i.e., a boost along the x-direction followed by a boost in the y-direction), we obtain a boost along ‘some’ direction given by Ω=tan−1[− βxγx βyγ y /(γx+γy)] in the x-y plane.                     + + + − ++ +− −− =Ω= − − 1000 0 1 )( 1 0 11 1 0 )( ˆˆ ˆˆˆ ˆˆ 2 ˆˆˆˆ ˆˆ ˆˆ 2 ˆˆˆˆ ˆˆ 2 ˆ 2 ˆ 2 ˆ ˆˆˆ ˆˆˆˆˆˆˆ ˆˆ 1 ˆ yx yxy yx yyxx yy xy yyxx yx yxx yxx yyyxxyx xyz γγ γγγ γγ γβγβ βγ γγ γβγβ γγ γβγ γβγ βγγβγγγ BBWBf mnmn ˆˆ 1 ˆˆ )( BBWB Ω= − ×××× Note that we can read BnBm =Wn××××m (Ω)Bf (e.g., By Bx =Wy××××x (Ω)Bf =W−z (Ω)Bf in the example above) backward to note that any boost B in the n-m (e.g., the x-y plane in the example above) can be decomposed into two mutually perpendicular boosts (in order) followed by a Wigner rotation (using group algebra – we mean it’s inverse)*: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ * Credit for this notation is based on Generic composition of boosts: An elementary derivation of the Wigner rotation, R. Ferraro and M. Thibeault, Eur. J. Phys. 20 (1999) 143-151. This result will be used later when we discuss particle representation in quantum field theory using Wigner basis states and especially when we calculate one first hand… ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 43
  44. 44. Since we will be using this soon let us look at a general example. Suppose that a parti- cle of mass mo is seen from system O with momentum p along the +z-axis. A second ob- server sees the same particle from a system O moving with velocity v along the +x-axis:             ΩΩ Ω−Ω−Ω+Ω−− Ω+ΩΩ−Ω =             ΩΩ Ω−Ω ΩΩ             − − =             ΩΩ Ω−Ω             + +             − − =ΛΛ=Λ − −−= cos0sin 0100 sincos0cossin sincos0cossin cos0sin 0100 sin0cos0 cos0sin 1000 0100 00 00 cos0sin0 0100 sin0cos0 0001 00 0100 0010 00 1000 0100 00 00 ),()()( ˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆ ˆˆˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ 1 ˆˆˆˆˆˆ zzzz xzzxxxzzxxzxx xxzzxxxzzxzx zzzz zzzzz xxx xxx zzz zzz xxx xxx yzxyzx γγγβ γγβγβγγβγβγγβ γβγβγγβγβγγγ γγγβ γβγβγ γγβ γβγ γγβ γβγ γγβ γβγ pWpLpL ×××× 2017 MRT        + −=Ω⇒ + −=Ω= Ω Ω Ω=Ω+−⇒Ω=Ω−Ω− zx zzxx zx zzxx zzxxzxzxzzxx ˆˆ ˆˆˆˆ ˆˆ ˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆˆˆ arctantan cos sin cossin)(sinsincos γγ γβγβ γγ γβγβ γβγβγγγγγβγβ Since Lx××××z(Λp) is symmetric we can extract the [Lx××××z(Λp)]3 2=[Lx××××z(Λp)]2 3 components:ˆ ˆ ˆ ˆ ˆ ˆ 44
  45. 45. px W−−−−y(Λ,p) Now, since: This provides us with the three Cartesian values for the boosted momentum: 2017 MRT y mo Suppose that observer O sees a particle (mass mo ≠0) with momentum pz in the z- direction. A second observer O moves relative to the first with velocity v in the x-direction. How does O describe the particle’s motion? Λp )ˆˆ(ˆˆ ˆˆˆˆˆoˆˆ kikip zzzxxzx γβγγβ +−=+−=Λ cmpp pz Λ(v) y x v ↑             −− − =                         − − =Λ zzz zzxxxzxx zzxxxzx zzz zzz xxx xxx zx ˆˆˆ ˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆˆ ˆˆ 00 0100 0 0 00 0100 0010 00 1000 0100 00 00 )( γγβ γβγβγγγβ γβγγβγγ γγβ γβγ γγβ γβγ pL             − =                         −− − =Λ cm cm cmcm kpL oˆˆ oˆˆˆ oˆˆo ˆˆˆ ˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆ ˆˆ 0 0 0 0 00 0100 0 0 )( zz zxx zx zzz zzxxxzxx zzxxxzx zx γβ γγβ γγ γγβ γβγβγγγβ γβγγβγγ µ we then have (N.B., with kµ=[moc,0,0,0]T a standard rest momentum for massive particles):          − =           Λ Λ Λ zz zxx ˆˆo ˆˆˆo 3 2 1 0 γβ γγβ cm cm p p p )ˆˆ(ˆˆ ˆˆˆˆˆooˆˆoˆˆˆ kikip zzzxxzzzxx γβγγβγβγγβ +−=+−=Λ cmcmcm x z z Lz(p) which when applied in the Figure will look like: 45
  46. 46. The Poincaré Algebra and is a space-time translation, i.e., as the product operation of a translation by a real vector aµ =[τ,a] and a homogeneous Lorentz transformation, Λµ ν (the translation being performed after the homogeneous Lorentz transformation.) It can conveniently be represented by the following matrix equation: The inhomogeneous Lorentz transformation (or Poincaré group), L={Λ,a}, is defined by: 2017 MRT                                 ΛΛΛΛ ΛΛΛΛ ΛΛΛΛ ΛΛΛΛ =                 1100001 3 2 1 0 33 3 3 2 3 1 3 0 22 3 2 2 2 1 2 0 11 3 1 2 1 1 1 0 00 3 0 2 0 1 0 0 3 2 1 0 x x x x a a a a x x x x The commutation rules of these generators with themselves are the Poincaré Algebra: µν ν µµµ axx +Λ== )( xL ρµσνρνσµµσµρµσνρρσµνρµ ηηηη MMMMMMiPP +−−== ],[0],[ and ρµσσµρσρµ ηη PPMPi −=],[ where the last coordinate, i.e., 1, has no physical significance and is left invariant by the transformation. The generators for infinitesimal translations are the Hermitian operators Pµ, and their commutative relations with the Hermitian generators for ‘rotations’ in the xµ -xν plane, Mµν =−Mν µ are as expressed in contravariant form: 46
  47. 47. Lorentz Transformations We note from (detΛ)2=1 above that either detΛ=+1 or detΛ=−1; those transformations with detΛ=+1 form a subgroup of either the homogeneous or the inhomogeneous Lorentz group. Furthermore, from the 00-components of ηµν Λµ ρ Λν σ=ηρσ and Λν σ Λκ τ ηστ =ηνκ, we have: These transformations also form a group. Now, those transformations with aµ =0 form a subgroup with to the Poincaré group: with i summed over the values 1, 2, and 3. We see that either Λ0 0≥+1 or Λ0 0≤−1. Those transformations with Λ0 0≥+1 form a subgroup. Note that if Λµ ν and Λµ ν are two such Λs, then: ii ii 00 00 2 0 0 11)( ΛΛ+=ΛΛ+=Λ 0 3 3 0 0 2 2 0 0 1 1 0 0 0 0 0 0 0 0 0 )( ΛΛ+ΛΛ+ΛΛ+ΛΛ=ΛΛ≡ΛΛ µ µ )0,()0,()0,( ΛΛ=ΛΛ TTT 2017 MRT Taking the determinant of ηµν Λµ ρ Λν σ=ηρσ gives: so Λµ ν has an inverse, [Λ−1]ν σ, which we see from ηµν Λµ ρ Λν σ=ηρσ and takes the form: σ µσρ µν ρ νν ρ ηη Λ=Λ=Λ− ][ 1 These transformations also form a group. known as the homogeneous Lorentz group. If we first perform a Lorentz transformation Λ: ν ν ρ ρ µρ ρ µµ xxx ΛΛ=Λ= 1)det( 2 =Λ 47
  48. 48. The problem of classifying all the irreducible unitary representations of the inhomo- geneous Lorentz group (i.e., the Poincaré group) can again be formulated in terms of finding all the representations of the commutation rules above by self-adjoint operators. commute with all the infinitesimal generators, Mµν and Pµ, and constitute the invariants of the group. 2017 MRT ν µνσ µσ σ σ µν µνµ µ µ µ PPMMPPMMWWWPPP −=== 2 122 and where Wµ is the Pauli-Lubanski vector: ρσν µνρσµ ε MPW 2 1 = They are therefore multiples of the identity for every irreducible representation of the inhomogeneous Lorentz group and their eigenvalues can be used to classify the irreducible representations. The set of all four-dimensional translations is a commutative subgroup of the inhomogeneous Lorentz group. Since it is commutative, the irreducible unitary representations of this subgroup are all one-dimensional and are obtained by exponentiation. Believe it or not, the following scalar operators: 48
  49. 49. and so: The subgroup of Lorentz transformations with detΛ=+1 and Λ0 0≥+1 is known as the proper orthochronous Lorentz group. 1)(1)( 2 0 02 0 0 0 3 3 0 0 2 2 0 0 1 1 0 0 0 −Λ−Λ≤ΛΛ+ΛΛ+ΛΛ≡ΛΛ i i But (Λ0 0)2=1+Λi 0Λi 0=1+Λ0 iΛ0 i shows that the three-vector [Λ1 0,Λ2 0,Λ3 0] has length √[(Λ0 0)2−1], and similarly the three-vector [Λ0 1,Λ0 2,Λ0 3] has length √[(Λ0 0)2−1], so the scalar product of these two three-vector is bounded by: 1)(1)()( 2 0 02 0 0 0 0 0 0 0 0 −Λ−Λ−ΛΛ≥ΛΛ 2017 MRT 49 If we look over the Lorentz transformation properties of Pρ and Mρσ is the case for homogeneous Lorentz transformations (i.e., aµ =0), we get: µνσ ν ρ µ σρ µρ µ ρ MaUMaU PaUPaU ΛΛ=ΛΛ Λ=ΛΛ − − ),(),( ),(),( 1 1 These transformation rules simply say that Mρσ is a tensor and Pρ is a vector. For pure translations (i.e., with Λµ ν =δ µ ν ), they tell us that Pρ is translation-invariant, but Mρσ is not.
  50. 50. The group of Lorentz transformations contains a subgroup which is isomorphic to the familiar three-dimensional rotation group. This subgroup consists of all Λµ ν of the form: 2017 MRT             = RR 0 01 Λ where ΛR(R1) and ΛR(R2) are spatial rotations and Λ(L1) a Lorentz transformation in the x1-direction. If we set µ =ν =0 in the equation Λλµ ηλρ Λρν =ηµν, we then obtain: so that Λ0 0≥1 or Λ0 0≤−1. where R is a 3×3 matrix with RRT =RTR=1. We call such a ΛR a ‘spatial’ rotation. Every homogeneous Lorentz transformation can be decomposed as follows: 1)()()(1)(1)( 2 0 32 0 22 0 1 3 1 2 0 2 0 0 ≥Λ+Λ+Λ+=Λ+=Λ ∑=i i )(Λ)(Λ)(ΛΛ 112 RLR RR= Remember that a Lorentz transformation for which Λ0 0≥1 is called an orthochronous Lorentz transformation. A Lorentz transformation is orthochronous if and only if it transforms every positive time-like vector into a positive time-like vector. The set of all orthochronous Lorentz transformations forms a group: the orthochronous Lorentz group. 50
  51. 51. The problem of finding the representation of the ‘restricted’ Lorentz group is equivalent to finding all the representations of the commutation rules above. The finite dimensional irreducible representation of the restricted group can be labeled by two discrete indices which can take on a values the positive integers, the positive half-integers, and zero. To show this, let us define the operators: 2017 MRT and their commutation rules are: From these operators, we can construct these invariants of the group: ( ) ( ) ( )Boost momentumAngular Momentum ],,[ ],,[ ],,[ 030201 211323 321 MMM MMM PPP = = = K J P kijkji kijkji kijkji KKJ JKK JJJ ε ε ε = −= = ],[ ],[ ],[ which commute with all the Ji and Ki. They are therefore the invariants of the group and they are multiples of the identity in anyirreduciblerepresentation.The representations can thus be labeled by the values of these operators in the given representation. and µν µν MM2 122 =− KJ ρσµν µνρσε MM8 1=•− KJ 51
  52. 52. where E(p)=ωp =√( p2c2 +mo 2c4) is the one-particle energy. As seen earlier, a rotation Rθθθθ by an angleθ =|θθθθ| about the direction of θθθθ=θp is representedontheHilbert space by: 2017 MRT ),(e),(e),(),( j aP i j aP i j mpmpmpaU ΨΨΨΨΨΨΨΨΨΨΨΨ µ µ µ µ hh −− ==1 hence: ),(e),()0,( j i j mpmpU ΨΨΨΨΨΨΨΨ θJ θ •− = h A boost K in the direction of the momentum ζζζζ: The operator corresponding to the translation by the four-vector aµ is given by: ),(e),()0,( j ci j mpmpU ΨΨΨΨΨΨΨΨ Kζ ζ •− = h         =         = −− cmcm i i o 1 o 1 sinˆsin p p p p p ζ and similarly for evolution: ),(e),(e ω j t i j tH i mpmp p ΨΨΨΨΨΨΨΨ hh = In an irreducible representation, the operation of translation by a thus corresponds to multiplying each basis vector |ΨΨΨΨ( p,mj)〉 by exp(−iPµaµ /h): µ µ aP i aU h − = e),(1 ˆ 52
  53. 53. which satisfy the following commutation rules: To make the range of values of the label more transparent, let us introduce the following generators: 2017 MRT )( 2 1 )( 2 1 iiiiii KiJiKKiJiJ −=+= and and It follows from the commutation rules that a finite dimensional irreducible representation space, Vjj' can be spanned by a set of (2j +1)(2j' +1) basis vectors | jmj, j'm'j〉 where j, mj, j' and m'j are integers or half-odd integers, −j ≤ mj ≤ j, −j' ≤ m'j ≤ j' and in terms of which the J and K operators have the following representation: 0],[ ],[ ],[ = = = ji kijkji kijkji KJ JiKK JiJJ ε ε jjjjj jjjjjjjj mjmjmmjmjJ mjmjmjmjmjmjJiJmjmjJ ′′=′′ ′′±+±=′′±=′′± ,;,,;, ,;1,)1)((,;,)(,;, 3 21 h hm jjjjj jjjjjjjj mjmjmmjmjK mjmjmjmjmjmjKiKmjmjK ′′′=′′ ±′′+′±′′′=′′±=′′± ,;,,;, 1,;,)1)((,;,)(,;, 3 21 h hm 53
  54. 54. These are thus a denumerable infinity of non equivalent finite-dimensional (in general they are non-unitary) irreducible representations. These can be labeled by two non-ne- gative indices ( j, j') where j, j' = 0, 1/2, 1, 3/2,…. The dimension of the representation is (2j +1)(2j' +1) and D( j, j') is single-valued if j+j' is an integer and double-valued. A quantity which transforms under D(0,0) is called a scalar, one which transforms under D(1/2,1/2) a four-component vector, one which transforms under the (1/2,0) representation a two-component spinor. A quantity which transforms under (0,1/2) is called a conjugate spinor. For the D(0,1/2) and the D(1/2,0) representations, an explicit matrix representation of the infinitesimal generators can be given in terms of the Pauli matrices with: 2017 MRT iiii iiii KK iJiJ σσ σσ 2 1 2 1 2 1 2 1 )2/1,0()0,2/1( )2/1,0()0,2/1( +=−= −=−= Note also that the quantity ξξξξ∗ ξξξξ is not a scalar. A two-component spinor, ξξξξ, transforms under spatial rotation as in the three- dimensional situation. For example, under an infinitesimal rotation ε about the i-th axis: and under an infinitesimal Lorentz transformation in the xi-direction, this spinor transforms according to: ( )3)½1( Ri i ξξξ σε+=→ ( )4Λ)½1( ξξξ εσi+=→ 54
  55. 55. ∑∫∑∫ == + = jj m jj j m jj P pd mpψmp P pd xψmpmpψ 0 * 12 1 0 ),(),()(),(),( φφφ ΨΨΨΨΨΨΨΨ Lorentz Invariant Scalar For the last case, Pµ = 0, the complete system of (infinite dimensional) unitary representations coincides with the complete system of (infinite dimensional) unitary representations of the homogeneous Lorentz group which we studied earlier. The irreducible representations of the inhomogeneous Lorentz group can now be classi- fied according to whether Pµ is space-like, time-like, or null vector, or Pµ is equal to zero. The representations of principle interest for physical applications are those for which P2 =mo 2c2 =positive constant, and those for which P2 =0. 2017 MRT Defining a Lorentz invariant scalar product within the vector space by integrating over a set of p (with P2 =mo 2c2, P0 =+√(p2 +mo 2c2)=E/c) and summing over the index mj: where j =0,1/2,1,3/2,2,…. So, we have hinted that an irreducible representation of the type P2 >0, P0 >0 is labeled by two indices (mo, j), where mo is a positive number and j is an integer or half-integer. The index mo characterizes the mass of the elementary system, the index j the angular momentum in its rest frame, i.e., the spin of the elementary system. The fact that the irreducible representation is infinite dimensional is just the expression of the fact that each elementary system is capable of assuming infinitely many linearly independent states. Let us first discuss the case P2 =mo 2c2. In that case, P0/|P0|, the sign of the energy, commutes with all the infinitesimal generators and is therefore an invariant of the group. 55
  56. 56. Klein-Gordon & Dirac For j = 0, the representation space is spanned by the positive energy solutions of the relativistically covariant equation for a spin-0 particle – the Klein-Gordon equation: For each (mo , j) – and a given sign of the energy – there is one and only one irreducible representation of the inhomogeneous Lorentz group to within unitary equivalence. For j half-integral, the representation is double-valued. 2017 MRT For j = 1/2 by the positive energy solution of the Dirac equation (free particle case): For j = 1 by the positive energy solutions of the Proca equation (not discussed). ),()( ),( 42 o 222 2 2 2 tcmc t t r r ϕ ϕ +∇−= ∂ ∂ − hh ),()(),( 2 o tcmciti rr ΨΨΨΨ∇∇∇∇ααααΨΨΨΨ β+•−= h&h We will discuss both the Klein-Gordon and Dirac equations later. )( 2 1 4 1)( 42 o µν µ ν µ µ ν ν µ µ ν µ φφφ φφφφφ xcm xxxxt x i         +         ∂ ∂ − ∂ ∂         ∂ ∂ − ∂ ∂ −= ∂ ∂ h 0)(2 o =         +∂− ∑ xcmc ψγ µ µ µ h )(),()()(),( tdmpxxψt j ΨΨΨΨΨΨΨΨΨΨΨΨ ⋅==≡ ∫ ∞+ ∞− ppprr ψµ or: such that in what follows,it applies also to states that get acted on by the Dirac equation: 56
  57. 57. One-Particle States The physical states of particles are described by the Wigner basis states |ΨΨΨΨkmo ( j,mj)〉 (which are equivalent to the states|ΨΨΨΨ(k,mj)〉) for a unitary irreducible representation of the inhomogeneous Lorentz group (Poincaré group) with Σµ pµ p µ =p02 −p2 =(moc)2. Thesestatesformthe Hilbert space of the theory and the momentum states |ΨΨΨΨ(p,mj)〉 can be obtained from the standard state |ΨΨΨΨpmo ( j,mj)〉≡|ΨΨΨΨ( p,mj)〉 by a unitary transformation: ∑ + −=′ Λ ′−•− ′Λ Λ =≡ j jm jEp m m j tE i j j j mjpW pE pE xx ),()],([e )( )( )π2( 1 )()( )( )( 2/3 ΨΨΨΨD rp h h ψψ µ Our goal is to find eigenkets of |ΨΨΨΨpmo ( j,mj)〉 as they appear following an homogeneous Lorentz transformation group U(ΛΛΛΛ,a) product on a state-vector |ΨΨΨΨpmo ( j,mj)〉 is as follows: where L( p) is some standard Lorentz transformationmatrix that depends on p=pµ and the momentum states are normalized over intermediate states andweget the coordinate ket: ),()(),( jj mkpLmp ΨΨΨΨΨΨΨΨ = ),()]([)]([)]([)]([ )( ),( ),()]([)]([ )( ),(),(),( o oo 1o o jmk jmkjmp mjpUpLUUpLU pE m aU mjpUU pE m aUmjaU ΨΨΨΨ ΨΨΨΨΨΨΨΨ LvΛ1 LvΛ1Λ ΛΛ= = − 2017 MRT The spin j corresponds to the eigenvaluesJ2= j( j +1)h2of J2 and J3 =mj h (mj = j, j −1,…,−j). Also, U(1,a)|ΨΨΨΨkmo ( j,mj)〉 meansthesamethingas exp(−iΣµ kµ aµ)|ΨΨΨΨkmo ( j,mj)〉(pµ =hkµ ). 57
  58. 58. The Lorentz transformations associated with the Poincaré group can be constructed in which the quantum state differs by only a mixture of the internal spin indices mj (i.e., run- ning over the discrete values j, j −1,…,−j) but possess the same physical observables. The effects of the inhomogeneous Lorentz transformations (e.g., acting on the Dirac fields) can be elucidated by considering the particle states obtained from the irreducible unitary representations of the Poincaré group. The unitary operation representing the Lorentz transformations acting on the Poincaré generators are given by the equations U(Λ,a)PρU−1(Λ,a)=Σµ[Λ−1]ρ µPµ & U(Λ,a)MρσU−1(Λ,a)=Σµν [Λ−1]ρ µ [Λ−1]σ ν (Mµν−aµ Pν −aν Pµ). Since the momentum four-vector commutes among each other according to [Pµ ,Pν]=0, the particle states can be characterized by the four-momentum Pµ together with additional internal degrees of freedom mj. The internal degrees of freedom pertain to the spin vector which can be affected by transformations in the space-time coordinates. and the state transforms accordingly for a space-time translation: ),(),( jj mppmpP ΨΨΨΨΨΨΨΨ µµ = ),(e),(e),(),( j ap i j aP i j mpmpmpaU ΨΨΨΨΨΨΨΨΨΨΨΨ ∑∑ −− =≡ µ µ µ µ µ µ hh1 Thus, the one-particle state is an eigenvector of the momentum operator: 2017 MRT The components of the energy-momentumfour-vector operator Pµ all commute with each other (i.e., [Pµ,Pν ]=0), so it is natural to express physical state-vectors in terms of eigenvectors of the four-momentum.This four-momentum Pµ is a trusted observable! 58
  59. 59. Under space-time translations, the states |ΨΨΨΨ(p,mj)〉 transforms as: 2017 MRT It is thus natural to identify the states of a specific particle type with the components of a representation of the inhomogeneous Lorentz group which is irreducible. Hence U(ΛΛΛΛ)|ΨΨΨΨ(p,mj)〉 must be a linear combination of the state vectors |ΨΨΨΨ(Λp,mj)〉: We must now consider how these states transform under homogeneous Lorentz transformations U(ΛΛΛΛ,0)≡U(ΛΛΛΛ) is to produce eigenvectors of the four-momentum with eigenvalues Λp (where we use U −1 (ΛΛΛΛ,a)Pµ U(ΛΛΛΛ,a)=Σν [Λ−1]µ ν Pν ): ),()()(),()( ),(][)(),()]()([)(),()( 11 jj jjj mpUpmpUp mpPUmpUPUUmpUP ΨΨΨΨΨΨΨΨ ΨΨΨΨΨΨΨΨΨΨΨΨ ΛΛ ΛΛΛΛΛ µ ν ν ν µ ν ν ν µµµ Λ=Λ=         Λ== ∑ ∑ −− ),(),( jj mppmpP ΨΨΨΨΨΨΨΨ µµ = We now introduce a label mj to denote all other degrees of freedom (i.e., all the other total angular momentum orientations), and thus consider state-vectors |ΨΨΨΨ(p,mj)〉 with: ),(e),(),( j ap i j mpmpaU ΨΨΨΨΨΨΨΨ ∑− = µ µ µ h1 ∑′ ′ ′ΛΛ= j j j m j m mj mppCmpU ),()],([),()( ΨΨΨΨΨΨΨΨΛ 59
  60. 60. In other words, if a system is confronted with a homogeneous Lorentz transformation ΛΛΛΛ, the momentum p is changed to Λp. According to Pµ |ΨΨΨΨ(p,mj)〉=pµ |ΨΨΨΨ(p,mj)〉, the one- particle state must possess an eigenvalue of Λp as well: in which U(ΛΛΛΛ,a)Pµ U−1(ΛΛΛΛ,a)=Σν [Λ−1]ν µ Pν has been used for the Lorentz transformed momentum generator. It can be seen then that U(ΛΛΛΛ)|ΨΨΨΨ(p,mj)〉 is a linear combination of the states |ΨΨΨΨ(Λp,m′j)〉, where: and this, believe it or not, does leave the momenta of all the particle states invariant. ∑∑ ′ ′ + −=′ ′ ′ΛΛ≡′ΛΛ= j jj j j j m j j mm j jm j m m j j mppmppmpU ),(),(),()],([),()( )()( ΨΨΨΨΨΨΨΨΨΨΨΨ DDΛ ),()(),(),( 2 o jjj mpcmmpppmpPP ΨΨΨΨΨΨΨΨΨΨΨΨ == ∑∑ µ µ µ µ µ µ ),()()( ),(])[(),()]()()[(),()0,( 1 j jjj mpUp mpPUmpUPUUmpUP ΨΨΨΨ ΨΨΨΨΨΨΨΨΨΨΨΨ Λ ΛΛΛΛΛ µ ν ν ν µµµ Λ= Λ== ∑− in which Dm′jmj ( j) (Λ,p) is termed the Wigner coefficient and, as said previously, they depend on the irreducible representations of the Poincaré group. As a special example, the Casimir operator Σµ Pµ Pµ cannot change the value of the momentum (i.e., its an invariant – as also stated previously): 2017 MRT 60
  61. 61. Hence, to distinguish each state, the standard four-momentum given by kµ =[moc,0,0,0] is chosen, from which all momenta can be achieved by means of a pure Lorentz boost: which implies that there exists a subgroup of elements consisting of some arbitrary Wigner rotations, W, and this subgroup is called the little group. µ ν ν ν µ kkW =∑ Notice that a simple three-dimensional rotation, W (which is an element of the Poincaré group), will render the standard four-momentum invariant: It is important to distinguish that this little group is not unique in the Lorentz group but it is actually isomorphic to other subgroups under a similarity transformation. This is because there is no well-defined frame for the standard momentum kµ due to the equivalence principle in special relativity.Thedefinitionofthelittle group is dependent on the choice of the standard momentum as well as the Lorentz transformation Λ. Wigner’s Little Group (N.B., the standard four-momentum is non-unique and it also depends on the charac- teristics of the particle, e.g., whether it is a massive or a massless particle). kpLp )(= 2017 MRT ),(),(),(),(),( 2 o jjjjjj mkmmkmkmkcmmkH ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ h=== J0p and& In this momentum rest frame, the state |ΨΨΨΨ(k,mj)〉 is specified in terms of the eigen- values of the Hamiltonian H=p0, the momentum p, and the z-component of the total angular momentum operator J as: 61
  62. 62. When L(p) is applied to k we get the momentum p. When ΛΛΛΛ is applied next to p it becomes Λp. When L−1(Λp) is applied to Λp we recover k! So, the only functions of pµ that are left invariant by all proper orthochronous Lorentz transformations Λµ ν are the invariant square p2 =Σµν ηµν pµ pν, and p2 ≤ 0, also the sign of p0. Hence, for each value of p2, and (e.g., for p2 ≤0) each sign of p0, we can choose a ‘standard’ four-momentum, kµ, and express any pµ of this class as: where Lµ ν is some standard Lorentz transformation that dependsonpµ,andalso implicitly on our choice of the standard kµ. We can define the states |ΨΨΨΨ(p,mj)〉 of momentum p by: ∑= ν ν ν µµ kpLp )( Wigner’s little group is the Lorentz transformation L−1(Λp) ΛΛΛΛL(p) that takes k to L(p)k =p, and then to Λp, and then back to k, so it belongs to the subgroup of the homogeneous Lorentz group. Operating on this equation with an arbitrary homogeneous Lorentz transformation U(ΛΛΛΛ), we find: k µ =[moc,0,0,0] pµ =Σν Lµ ν ( p)kνΛ p=Σν Λ0 ν pν L−1(Λ p) L(p) ΛΛΛΛ ),()]([)(),( jj mkpUpNmp ΨΨΨΨΨΨΨΨ L= N(p) is a numerical normalization factor to be chosen on the next slideandwhere U[L(p)] is a unitary operator associated with the pure Lorentz ‘boost’ thattakes[moc,0] into[p0,p]. The transformation Wµ ν (i.e., L−1(Λp) ΛΛΛΛL(p) = W(Λ, p)) thus leaves k µ invariant: Σν Wµ ν kν =kµ. For any Wµ ν satisfying this relationship, we have: ),()]()([)]([)( ),()]([)(),()( 1 1 j jj mkppUpUpN mkpUpNmpU ΨΨΨΨ ΨΨΨΨΨΨΨΨ LΛLL LΛΛ 4444 34444 21 = − ΛΛ= = ∑ + −=′ ′ ′Λ=Λ j jm j j mmj j jj mkpmkpU ),()],([),()],([ )( ΨΨΨΨΨΨΨΨ WW D where the coefficients Dm′j mj ( j ) [W(Λ, p )] furnish the representation of the little group. 2017 MRT 62
  63. 63. The Wigner coefficient for the standard four-momentum can be evaluated as: which implies directly that: )()()( WWWW UUU ≡ Note that the Wigner coefficients form a representation of the little group, which means that for any little group elements W, W, the group multiplication property holds: ∑∑ ′ ′ + −=′ ′ ′≡′= m mm j jm j j mmj mkWmkkWmpU j jj ),()(),(),(),()( )( ΨΨΨΨΨΨΨΨΨΨΨΨ DDW thus: ∑ ∑ ′ ′′ ′ ′ ′′= ≡′′= σ σσ σ σσ σ σσ , ),()()( ),()()(),()(),()( m mm kWW kUUkWWmkU ΨΨΨΨ ΨΨΨΨΨΨΨΨΨΨΨΨ DD D WWWW ∑ ′=′ ′′ m mm WWWW )()()( σσσσ DDD However, this choice for the normalization condition leads to problems with the subsequent transformation equations relating to the particle states which involve some tedious momentum-dependent constants. 2017 MRT 63
  64. 64. Normalization Factor Using W=L−1 ΛΛΛΛL and inserting into U(ΛΛΛΛ)|ΨΨΨΨ(p,m)〉=NU[L(Λp)]U[L−1(Λp)ΛΛΛΛL(p)]|ΨΨΨΨ(p,m)〉 and using U(W)|ΨΨΨΨ(k,m)〉=Σm′ Dm′ m(W )|ΨΨΨΨ(k,m′)〉 we get: or, recalling the definition |ΨΨΨΨ〉=NU[L(p)]|ΨΨΨΨ〉, we finally get: 2017 MRT for which The normalization factor N(p) is sometimes chosen to be N(p)=1 but then we would need to keep track of the p0/k0 factor in scalar products. Instead, the convention is that: The normalization condition is achieved using the scalar product: ∑′ ′ ′ΛΛ Λ = m mm mppW pN pN mpU ),()],([ )( )( ),()( ΨΨΨΨΨΨΨΨ DΛ ∑′ ′ ′ΛΛ= m mm mkpUpWpNmpU ),()]([)],([)(),()( ΨΨΨΨΨΨΨΨ LΛ D 0 0 )( p k pN = mmmpmp ′−′=′′ δδ )(),(),( ppΨΨΨΨΨΨΨΨ )()(),(),( 2 kk −′=′′ ′ δδ mmpNmpmp ΨΨΨΨΨΨΨΨ )( )()( )( )( )( )( 0 0 0 0 pE pE p p pN pN p k pN Λ = Λ = Λ ⇒ Λ =Λ and 64
  65. 65. We choose the following normalization condition (c.f., Weinberg QTF I): which implies that: This implies that the Wigner coefficients cannot sum up to unity, but instead up to a phase factor that depends on the momentum of the particle, p, and the Lorentz transformation,ΛΛΛΛ: mmpmkmp ′′−=′ δδ )()2(),(),( )3(0 ppΨΨΨΨΨΨΨΨ σσσσ δ ′′ Λ =ΛΛ∑ 0 0 * )( ),(),( p p pp m mm DD mm mm mm mm mm pppppp mpmppppUUp ′ ′ ′′′ ′ ′′ ∑ ∑ ΛΛΛ==′ ′ΛΛΛΛ=′ δδσσ σσ σσσσ σσ , 0*0 , *† )2(),(),(2),(),( ),(),(),(),(),()()(),( DD DD ΨΨΨΨΨΨΨΨ ΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨΨ ΛΛ and finally: 10 0 † )( ),(),( p p pp Λ =ΛΛ DD The transformations mentionned above whereby the representations of the Poincaré group is being derived from the little group is termed as the method of induced representations. 2017 MRT 65
  66. 66. The explicit form for W(Λ,p) is: where the angle θ is defined by tanhθ =√[(p0)2 −mo 2c4]/p0 and the components of the particle momentum are given by pi (i=1,2,3). The Lorentz boost L(p) transforms the standard momentum kµ to momentum pµ. In addition, the general Lorentz transformation ΛL for ‘pure boost’ only is written as: ( )ppkpLpLpW pL Λ→ →ΛΛ=Λ Λ− )(1 )(),(),(         −+ = ji ji i j ppp p pL ˆˆ)1(coshsinhˆ sinhˆcosh )( θδθ θθ         −+ =Λ ji ji i j L nnn n ˆˆ)1(coshsinhˆ sinhˆcosh ξδξ ξξ and it can be shown that this is an element of the little group associated with the standard vector kµ. (N.B., L(Λp) is the Lorentz boost of the momentum (Λp)µ from standard momentum vector kµ ). Then the Wigner transformation corresponding to the present choice of kµ is a SO(3) rotation. Furthermore, the Lorentz boost L(p) can be computed from the boost generator Ki =Mi0=ih(xi ∂0 −x0∂i) where L(p)=exp(−icθ p•K/h). The explicit form of L(p) is given by: ˆ where n is a unit vector along the direction of the boost. This transforms the momentum pµ to a Lorentz-transformed momentum (Λp)µ. ˆ 2017 MRT 66 One-Particle States – con’d
  67. 67. Considering now a Taylor expansion of the Wigner transformationW(Λ,p)= L−1(Λp)ΛL(p) as well as the explicit form of the matrix for L(p) just obtained for an arbitrary Lorentz transformation Λ: in which the Lorentz boosts, L(p) and L(Λ,p) and the arbitrary Lorentz transformation, Λ, are parametrized in terms of its generators. In addition, the spatial component of the Lorentz-transformed momentum is given by pΛ=Λp (N.B., The corresponding Lorentz boost for the Lorentz transformed momentum starting from the standard vector, kµ, is given by tanhθ′=|pΛ|/(Λp)0, in which L(Λp)k=Λp). K+−=Λ µν µν Mω 2 1 )ω( 1 thus: KpKp •−−•′ − ≡ ΛΛ=Λ ˆω 2 1 ˆ 1 eee )()(),( Λ θθ µν µν hh ci M ci pLpLpW βα β αβα βµν µναα pppM i pp ]ω[]ω[ 2 )( +≡−≅Λ Consider the inifinitesimal Lorentz transformation being parametrized by the anti- symmetric tensor ωµν (expressed to first order terms of ω): 2017 MRT Note that in the above and from now on, we restored the summation convention where we sum over any space-time index which appears twice in the same term. 67
  68. 68. A Taylor series expansion of the Wigner transformation is considered as fallows: thus: In the ‘second term’ of this last equation the expression can be re-written as: K K + ∂ ΛΛ∂ += + ∂ Λ∂ +Λ=Λ = − = = 0ω 1 0ω 0ω ω )]()([ ω ω ),( ω),(),( µν µν µν µν pLpL pW pWpW 1 K K +            −Λ+         ∂ Λ∂ += +         ∂ Λ∂ Λ+         Λ ∂ Λ∂ +≅Λ = − = − = − = − )( 2 )(ω)( ω )( ω )( ω )ω( )(ω)()ω( ω )( ω),( 0ω 1 0ω 1 0ω 1 0ω 1 pLM i pLpL pL pLpLpL pL pW µν µν µν µν µν µν µν µν 1 1 in which the substitution Λp=p′ has been made and, in general, L−1(p′)=ηηηηLT(p′)ηηηη. 2017 MRT 68                       −+               ′ ′′         − ′ + ′ − ′ − ′ ∂ ∂ +=         ∂ Λ∂ = = − 22 o 0 o o 2 o 0 0ω 22 o 0 o o 2 o 0 0ω 1 11 ω ω)( ω )( ω p pp cm p cm p cm p cm p p pp cm p cm p cm p cm p pL pL kj kj j k ji ji j i δδ µν µν µν µν 1
  69. 69. Now, since (Λp)α≅pα−(i/2)[ωµν Mµν]α β pβ≡pα+[ω]α β pβ, we get (h=c=1): and since ξ i =ω0i =−ωi0 and θ k =εijk ωij as well as Ji =εijk Mjk and Ki =Mi0: βα β βα βµν µν µν α µν ppM ip ]ω[][ω 2ω )( ω 0ω ≡−≅         ∂ Λ∂ = in which the axial vector is given by θθθθ=[θ 1,θ 2,θ 3]† and mi =(p××××θθθθ)i as well as:         − • ≡         − • =         − =                 − = +−=++−= ii ii k jkjii jj j k kjii j k ki i ji ji i i i i p ppp p p p pJK i pMMM i p mp ξpp θpp ξpp ξ ξθεξ ξ θεξ ξ θξ βββ β α 0 00 0 0 0 0 0 )ˆ( )ˆ( )ˆ(0 ]22[ 2 ]ωωω[ 2 ]ω[ ××××         −−−=      ∂ ′∂ ∑ = nm nnmiii i pppp p ][ˆˆ][ 1 ω )ˆ( ω 00 0ω mpmp p ξξµν µν At this junction, the sum Σmn pm[p0ξn −|p|mn] describes the relation between the components of the momentum and its Lorentz boost. ˆ 2017 MRT ˆ 69
  70. 70. Assuming that any sum over the cross-terms is zero for m≠n, that is: )ˆ(][ˆ 00 ξpmp •=−∑ ppp nm nnm ξ in which the boost vector is given by ξξξξ=[ξ 1,ξ 2,ξ 3]†. With the axial vector, θθθθ=[θ 1,θ 2,θ 3]†, θθθθ and ξξξξ become the parameter vectors associated with the infinitesimal Lorentz transformation Λ. In addition, the vector normal to both the axial and momentum vectors is given by m=p××××θθθθ. Jmpξp p Km p pξpξ pmξp p m p ξp m p ξp ξp p m p m p ξp p •         −         −+•         −•         −−≡                       −         −−+•         − −+•         − =                       −+                       •         − −−         −+− +−• =         ∂ Λ∂ = − ))))×××××××× ×××××××× ˆ(2)ˆ(1ˆ)ˆ(1 )ˆ(2)ˆ(1ˆ)ˆ(1 ˆ)ˆ(10 ˆˆ1ˆˆ)ˆ(]ω[ˆ]ω[ˆ1 )ˆ( )( ω )( ω 0 o 0 oo 0 o 0 0 o 0 o 0 oo 0 o 0 oo 0 o 0 o oo 0 o 0 o 0 oo 0 oo 0 o 0ω 1 p m p i mm p m p i p m p m p m p m p m p m p m p pp m p m p m p m p pp mp pppp m p mm p mm p m pL pL nkinniii kkk kj kj j k jiijjiii jj εξ ξ δξ ξ β β β β µν µν ˆ Our equation for ωµν [∂L−1(Λp)/∂ωµν ]|ω=0L(p) above (i.e. the ‘second term’) can be simplified by using the Lorentz boosts from the matrices for L(p) and ΛL further up, and this can be worked out as: 2017 MRT 70
  71. 71. The third term in our Taylor expansion for W(Λ,p) above (i.e., the term obtained above ωµν [L−1(Λp)(−½i Mµν )]|ω=0L(p)) can also be simplified in terms of its matrix elements (Note that the expression (−½i ωµν Mµν ) can be re-written as the matrix [ω]): Jmpθξp p Km p pξpξ mp p ξm p ξp m p ξp m p ξp m p ξp •                 −+−−+•         −•         −−−≡                                 −+−        −•         − − −+•         − =               −         − +−         −−•         − − −•         − − =                       −+        −               ′′         −+ ′ − ′ − ′ =′− ))))×××××××× ))))×××××××× ˆ(1)ˆ( 2 ˆ)ˆ(1 ˆ(1ˆ)ˆ( ˆ)ˆ(10 )ˆˆˆˆ(ˆ)ˆ( )(ˆ)ˆ(0 ˆˆ1 0 ˆˆ1 )](][ω)][([ o 00 oo 0 o 0 o 0 ooo o 0 o 0 o 0 oo 0 o o 0 oooo o 0 o 0 oo o 0 o 0 o 0 o oo 0 o 0 o oo 0 1 m pp i mm p m p i m p mm p m mp m p m p m p m p pmmp m mp m p m p m p m mp m p m p m mp m p pp m p m p m p m p pp m p m p m p m p pLpL mlimm m iii kkk lilimmlil il iiii lll lk lk k l mmkjj k ji ji i j εθξ ξ θεξξξ ξ δ θεξ ξ δ in which ωµν [∂( p′)α/∂ωµν]|ω=0 above has been used for the matrix reprentation of the arbitrary infinitesimal Lorentz transformation Λ(ω). 2017 MRT 71
  72. 72. After some careful re-grouping of the terms from all the matrix elements given in ωµν [∂L−1(Λp)/∂ωµν]|ω=0L(p) and [L−1(p′)][ω][L(p)] just obtained, the boost terms in our expansion for W(Λ,p) has cancelled out completely and leaving behind the rotation terms: where Ω( p) k =−ε ijk [ωij −(piω0j −pj ω0i)/( p0 +mo)] is the Wigner angle and Jk =½ε ijk Mij is the rotation generator for the Poincaré group. Thus, we obtain: J1 1 1 Jθξp p 1 •+=         − − −+=         −− − −= •         + − −≡Λ )( )ωω( )( 1 ω )( )( 1 )( )( ),( 00 o 0 o 0 o 0 pi Mpp mp i Mpp mp i mp ipW ji ijjiji jin njiijji n θεξξ ××××                   − − −+=Λ njinnmppW εθ)ˆ( )( 0 00 )],([ o 0 ξp p 1 ×××× 2017 MRT 72
  73. 73. The Wigner angle can be re-written as a sum of the contributions from the rotation and the boost: Here the angle of rotation are represented by the Euler angles θk =ε ijk ωij and the boost parameter ξi =ξ ni =ω0i. The finite Wigner transformation is given by: )ˆˆ( )( o 0 o 0 pn p θ ξp θ ×××× ×××× mp mp p + −≡ + −≡ ξ J• ∞→ =                     Λ=Λ i N N p N WpW e , ω lim]),ω([ ˆ 2017 MRT 73
  74. 74. The representation matrix Dmjσ ( j) [W(Λ,p)] can be constructed from the angular momentum generators Ji explicitly, depending on the angular momentum of the particle. For example, spin-1/2 particles will be considered to have appropriate generators as given by J=½σσσσ according to the isomorphism between the proper Lorentz group and the SU(2)⊗SU(2) algebra. J• =Λ i pW e),( The Wigner transformation corresponding to an arbitrary Lorentz transformation Λ is given by: where Ω( p)k =−½ε ijk [(piωj 0 −pj ωi 0)/(p0 +mo)] and Jk =½ε ijk Mjk in the absence of rotation. This can also be written as: o 0 o 0 ˆ )( mpmp p + = + −≡ pnξp ×××××××× ξ Here the boost parameters are represented by ξi =ξni =ωi 0. For an infinitesimal variation ωµν, the transformation matrix is (e.g., spin-1/2 case described above): ˆ                 + − +        •+                 + −=                 −         •+                 −≅                •+        =Λ × ×× 3 o 0 o 0 2 o 022 32 2222 )2/1( )(2 ˆ !3 1 )(2 ˆ )(2 ˆ !2 1 1 2!3 1 22!2 1 1 2 sin 2 cos)],([ mpmp i mp I iIiIpW p pnpn σ pn σσ ×××××××××××× ξξξ D 2017 MRT 74
  75. 75. The representation matrix becomes (i.e., spin-1/2 case): For the sake of convenience, I2×2 is assigned as the 2×2 identity matrix, and σσσσ as the ‘vector’ that is comprized of the Pauli 2×2 matrices [σ1,σ2,σ3]. (Note that the Wigner angle |ΩΩΩΩ|=Ω is dependent on both the rotation and boost parameters for an arbitrary Lorentz transformation Λ. In addition, notice that an additional normalization factor of √[p0/(Λp)0] has been appended to the matrix so as to maintain the condition given D†(Λ,p) D(Λ,p)=p0/(Λp)0 I2×2). by using the well known Pauli spin vector σσσσ relation (σσσσ•a)(σσσσ•b)=(a•b)+iσσσσ•(a××××b) and remembering that ΩΩΩΩ=ΩΩΩΩ /|ΩΩΩΩ|. Thus, with the well known sinξ=ξ−ξ 3/3!+ξ 5/5!−… and cosξ =1−ξ 2/2!+ξ 4/4!−… identities:                 •+        Λ =Λ 2 sin)ˆ( 2 cos )( )],([ 0 0 )2/1( σ1 i p p pWD                 +         −•+         +         − Λ =         +      •+      •+ Λ = Λ =Λ × × • KK K 32 220 0 2 220 0 2 0 0 )2/1( 2!3 1 2 )ˆ( 2!2 1 1 )( 2!2 1 2!1 1 )( e )( )],([ σ σσ σ iI p p iiI p p p p pW i D 2017 MRT ˆ 75
  76. 76. In the absence of rotation, the arbitrary Lorentz boost can be parametrized according to the matrix defined by ΛL above. We can construct the matrix by the associative property of the representation matrix: in which: )]([][)]([ )]()([)],([ )2/1()2/1(1)2/1( 1)2/1()2/1( pLpL pLpLpW DDD DD ΛΛ≡ ΛΛ=Λ − − The representation matrix D(1/2)[W(Λ,p)], in the absence of rotation, is this given by:             Ω •+      Ω Λ =             •−      •+      + +Λ+ Λ =Λ × ×× 2 sin)ˆ( 2 cos )( 2 sinh)]ˆ([ 2 sinh)ˆ( 2 cosh)( ]))[(( )/( )],([ 220 0 22o 0 22 o 0 o 0 00 )2/1( mσ npσnp iI p p iImpI mpmp pp pW ξξξ ××××D )ˆˆ(sinhsinh 2 1 coshcosh 2 1 2 1 )ˆˆ( 2 sinh 2 sinh ˆ 2 sin )ˆˆ(sinhsinh 2 1 coshcosh 2 1 2 1 )ˆˆ( 2 sinh 2 sinh 2 cosh 2 cosh 2 cos pn pn m pn pn •++             =      •++ •            +            =      θξθξ θξ θξθξ θξθξ ×××× & where coshθ =p0/mo and m=n××××p represents the axis of rotation of the equivalent Wigner transformation (e.g., of the Dirac spinors in the spin-½ case). Here the parameters ξ and n are defined in the same manner as the general Lorentz boost as given by the ΛL matrix. ˆ ˆˆ 2017 MRT ˆ 76
  77. 77. ∑−=′ ′ ′ΛΛ Λ =Λ j jm j j mmj j jj mppW p p mpU ),()],([ )( ),()( )( 0 0 ΨΨΨΨΨΨΨΨ D With the normalization and the understanding of D, our transformation thus becomes: where Wα µ (Λ,p)=(L−1)α σ (Λp)⊗ Λρ ν Lν µ (p) is theWigner Rotation,k≡kµ =[k0 =moc,ki =0], p≡pν =[p0 =E/c,pi ], and since U(W)|ΨΨΨΨ(k,mj)〉=Σm′j Dm′jmj ( j) [W(Λ,p)]|ΨΨΨΨ(k,m′j)〉: 2017 MRT since N(Λp)U[L(Λp)]|ΨΨΨΨ(k,m′j)〉=|ΨΨΨΨ(Λp,m′j)〉⇒U[L(Λp)]|ΨΨΨΨ(k,m′j)〉=N−1(Λp)|ΨΨΨΨ(Λp,m′j)〉 where the normalization constant is N(Λp)=√[k0/(Λp)0]. Finally we have: ),()],([)]([),(])()([)]([),()( 0 0 ),( 1 0 0 jj p j mkpUpLU p k mkpLpLUpLU p k mpU ΨΨΨΨΨΨΨΨΨΨΨΨ ΛΛ=ΛΛ= Λ − WΛΛ W 44 344 21 444 3444 21 ),( )( 0 0 )( 0 0 ),()()],([ ),()],([)(),()( j j jj j jj mp j m j mm j m j mmj mkpLpW p k mkpWpL p k mpU ′Λ ′ ′ ′ ′ ′ΛΛ= ′ΛΛ=Λ ∑ ∑ ΨΨΨΨ ΨΨΨΨ ΨΨΨΨΨΨΨΨ D D 77 Isn’t this the most beautiful equation you’ve ever seen? So, this is how particle states |ΨΨΨΨ( p,mj)〉 transform into when a Lorentz transformation Λ acts on it! That is, using a quantum mechanical unitary operator U(Λ) that changes momentum p and spin mj! 6447481 64748p = L(p)k
  78. 78. Mass Positive-Definite To define the mass positive-definite, we must recall a few definitions such as the unitary quantum transformation for coordinate and angular momenta: where P={P1,P2,P3}, J={M23,M31,M12} and K={M10,M20,M30} and their properties are P ρ† =P ρ , M ρσ † =M ρσ and M ρσ =−M σρ . Now for the definition of invariance under a unitary transformation of a homogeneous Lorentz Transformation Λ and a uniform translation a: ...ω 2 1 ),ω( ++−=+ ρσ ρσ ρ ρεε M i PU h 11 ),(),(),(),ω1(),( 1 jj mpmpaUUaU ΨΨΨΨΨΨΨΨ =Λ+Λ − ε which also applies to both momenta: )(),(),(),(),( 11 µννµµνσ ν ρ µ µνµρ µ µ PaPaMaUMaUPaUPaU +−ΛΛ=ΛΛΛ=ΛΛ −− & For the momentum, they commute with each other: 0],[ =νµ PP but for the angular momentum they do not: ρµσσµρρσµ ηη PPMPi −=],[ nor do the angular momenta together commute: ρµσνρνσµνσµρµσνρρσµν ηηηη MMMMMMi +−−=],[ As comparison, we show the case when Λ=1 then a= 0 and a rotation around the 3-axis: θµ µ pJ ˆ 3 e)0,(e),1( •−− == hh i aP i RUaU and ( ))()0,( Λ=Λ UU 2017 MRT 78
  79. 79. j i jj i jki j mm mjmjmjmjR jj ,e,,e,)]([ ˆ )( Ω•−•− ′ ′=′=Ω pJJ hhD ∑∑ ′ ′ ′ ′Ω=′Ω′=Ω j jj j m j j mm m jjjj mRmmjRmjmjR )]([,)]([,,)]([ )( DDD θϕ θϕ j jjjj j jj mi kmmkmmj mi k jjjj jjjjkj mm kmmkmjkmjk mjmjmjmj e 2 sin 2 cose )!()!()!(! )!()!()!()!( )1(),,( 222 )( +−′−′−+ ′       Ω       Ω +−′−′−−+ ′−′+−+ −=Ω ∑D ∑∑∑ Ω=′⇒Ω= ′ l ll l l l l l ll llll m mm m mmmmR zpzp ˆ,),,(ˆ,ˆ,,)]([ˆ )( ϕθDD 2017 MRT Recall also that your mathematical toolbox consists of these fundamental relations: and with Finally: jjjjjj mm j kikimm j mm M i ′′′ Ω+=Ω+ ][ 2 )( )()( δ1D In this instance, we take (i.e., Weinberg’s QFT definition – he also uses Θ for our Ω): where Rik =δik +Ωik with Ωik =−Ωki infinitesimal and for mj running over j, j−1,…, −j. Also: jjjjjj jjjjjj mmjmm j mm j z mmjjmm jj mm jj mJMJ mjmjJiJMiMJ ′′′ ±′′′± ==⇒ +±=±=±⇒ δ δ h hm )()( )1)(()()( )( 3 )( 12 1 )( 2 )( 1 )( 31 )( 23 79
  80. 80. Now, let us expand |ΨΨΨΨ(Λp; j,mj)〉=|ΨΨΨΨ(Λp)〉|l,s; j,mj 〉 in the case of an electron (s=±½): 2017 MRT ∑ +−−++−−−−+−−+ −+−+−+− × +++ +−+−+−+ +=≡ k ss jjss k sjjs j mm kmsjkmjkmskmkjsk mjmjmsmsmm sj jsjsjjs mmmmjsmmss )!()!()!()!()!(! )!()!()!()!()!()!()1( )!1( )12()!()!()!( ),(,;,,;, ll ll ll lll ll l lll lll δC Using Jz|l,s;j,mj 〉=mjh|l,s;j,mj 〉 we found out that that |l,s;j,mj 〉 is an eigenfunction of Jz and since Jz commutes with J2 the eigenfunction of Jz is simultaneously an eigenfunction of J2 which means that |l,s;ml,ms 〉 is not an eigenfunction of J2. To get around this, we constructed a linear combination of |l,s;ml,ms 〉 instead: such that |l,s; j,mj 〉 is simultaneously an eigenfunction of Jz and J2. The quantities Cj mlms = 〈l,s;ml,ms |l,s; j,mj 〉 are numerical coefficients which are known as Clebsch-Gordan (CG) coefficients – and for our benefit j=l+s and mj =ml +ms. A general formula for these coefficients is due to Wigner: with l=0,1,2,…, s=±½, j =0,1/2,1,3/2,…, ml =l,l−1,…,−l, ms =±s & mj = j, j −1,…,−j. ∑ ∑′ ′ Λ′Λ Λ =ΛΛ j s sj m mm sj j mmmsj pmmsmjspWmms p p mjpU l l ll lll )(,;,,;,)],([,;, )( ),;()( )( )(0 0 ΨΨΨΨΨΨΨΨ D ∑∑ ±= += ±= == )½( ½ ,;,,;,,;,,;,,;, s mmm s j mm mm jssj sj s s mmsmjsmmsmmsmjs l l l l l ll lllll C 80
  81. 81. a•+ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ ∇∇∇∇ t a z a y a x a t a a tzyxt µµ Point P at the tip of the distance vector xµ ={ct,r} is given in 4-dimensional Rectangular (Cartesian) Coordinates by the intersection of constant x, constant y and constant z planes and time t. The speed of light c is a constant of motion (the same everywhere!)       ∂ ∂ =      ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ =∇ φ φφφφφ φµ ∇∇∇∇,,,, tzyxt x r O The Laplacian ∇∇∇∇• ∇∇∇∇≡∇2 =∂2/∂x2+∂2/∂y2 +∂2/∂z2 leads to the D’Alembertian: The divergence ∇∇∇∇• A=∂Ax /∂x+∂Ay/∂y +∂Az/∂z gives: P 2017 MRT A ],[],,,[ atzyxt aaaaaa ==µ y φ z If the scalar product is A•B =Ax Bx +Ay By +Az Bz, then: ba•−=−−−= ttzzyyxxtt babababababa µµ       − ∂ ∂ =      ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ ∇ ∇∇∇∇,,,, tzyxt µ The gradient ∇∇∇∇ψ =∂ψ/∂xi +∂ψ/∂yj+ ∂ψ/∂zk of a scalar function ψ : ˆ ˆ ˆ Del ∇∇∇∇=∂/∂xi+∂/∂yj+∂/∂zk =[∂/∂x,∂/∂y,∂/∂z] is now:ˆ ˆ ˆ τ = ct Boosts & Rotations Electrodynamics provides the differential equations for the potentials: φ =ρ /εo and A= j/εo and the continuity equation: ∂φ/∂t + ∇∇∇∇•A=0. In this new in four-dimensional notation: Aµ ={φ,A} we get Aµ = jµ/εo and ∇µ Aµ =0. y z x y z x x0 = ct v The transformation laws which give φ and A in a moving system in terms of φ and A in a stationary system. Since Aµ ={φ,A} is a four-vector, the equation must just look like t= γ(t −|v|z/c2) and z= γ(z −|v|t) with γ =(1 −|v|2/c2)−1/2 except that t is replaced by φ, and r is replaced by A. Thus: v 22 2 )(1)(1 c A AAAAA c cA z zxxyy z v v v v − − === − − = φφ φ ,,, If the vector A = Axi + Ay j+ Azk =[ Ax , Ay , Az ], then we have the definition of the 4-vector aµ : ˆ ˆ ˆ =∇− ∂ ∂ = ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ =∇∇ 2 2 2 2 2 2 2 2 2 2 2 tzyxt µµ So = (1/c2)∂t−∇2 =ηµν ∂µ∂ν =∂µ∂µ with ∂µ =∂/∂µ. 81
  82. 82. ρσ ρσ ρ ρ M i P i UaU ω 2 1 ε)εω,(),( hh +−=+=Λ 11 ]ωinTerms[)ωω()ωδ)(ωδ( 2 O+++=++=ΛΛ= ρσσρσρσ ν σ ν ρ µ ρ µ µνσ ν ρ µ µνρσ ηηηη µµ ν µ ν µ ν µ εω =+=Λ aδ and               − − =                           −+− − =Λ==             )( )( ˆˆ)1(100 0100 0010 00 2 ctz y x zt z y x t c c xx z y x t v v vvv v γ γ γγ γγ νµ ν µ The Lorentz transformation transformsframe xν (say x,y,z,t) intoframe x µ (say x, y,z,t): assuming the inertial frameis going in the x3 (+z) direction.If both ω µ ν and εµ are taken to be an infinitesimal Lorentz transformation and an infinitesimal translation, respectively: This allows us to study the transformation: The linear unitary operator U=1+iε T was constructed: where the generators G of translation (Pµ ) and rotation (Mµν ) are given by P1, P2, and P3 (the components of the momentum operator), M23, M31, and M12 (the components of the angular momentum vector), and P0 is the energy operator. 2017 MRT x2 = y x3 = z x1 = x x2 = y x3 = z x1 = x x0 = ct β =|v|/c 82

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