Home   Search    Collections   Journals   About   Contact us    My IOPscience       Gauge systems and functions, hermitian...
XIV Mexican School on Particles and Fields                                                              IOP PublishingJour...
XIV Mexican School on Particles and Fields                                                            IOP PublishingJourna...
XIV Mexican School on Particles and Fields                                                           IOP PublishingJournal...
XIV Mexican School on Particles and Fields                                                                       IOP Publi...
XIV Mexican School on Particles and Fields                                                       IOP PublishingJournal of ...
XIV Mexican School on Particles and Fields                                                            IOP PublishingJourna...
Upcoming SlideShare
Loading in …5
×

Gauge systems and functions, hermitian operators and clocks as conjugate functions for the constraints.

342 views

Published on

Proceedings for the XIV Mexican School on Particles and Fields

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
342
On SlideShare
0
From Embeds
0
Number of Embeds
51
Actions
Shares
0
Downloads
1
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Gauge systems and functions, hermitian operators and clocks as conjugate functions for the constraints.

  1. 1. Home Search Collections Journals About Contact us My IOPscience Gauge systems and functions, hermitian operators and clocks as conjugate functions for the constraints This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Phys.: Conf. Ser. 287 012043 (http://iopscience.iop.org/1742-6596/287/1/012043) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.248.181.55 The article was downloaded on 20/05/2011 at 16:45 Please note that terms and conditions apply.
  2. 2. XIV Mexican School on Particles and Fields IOP PublishingJournal of Physics: Conference Series 287 (2011) 012043 doi:10.1088/1742-6596/287/1/012043Gauge systems and functions, hermitian operators andclocks as conjugate functions for the constraints Vladimir Cuesta 1,† , Jos´ David Vergara 1,†† and Merced Montesinos e 2,♦ 1 Instituto de Ciencias Nucleares, Universidad Nacional Aut´ noma de M´ xico, 70-543, Ciudad de o e M´ xico, M´ xico, e e 2 Departamento de F´sica, Centro de Investigaci´ n y de Estudios Avanzados del Instituto Polit´ cnico ı o e Nacional, Instituto Polit´ cnico Nacional 2508, San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad e de M´ xico, M´ xico e e E-mail: 1,† vladimir.cuesta@nucleares.unam.mx 1,†† vergara@nucleares.unam.mx 2,♦ merced@fis.cinvestav.mx Abstract. We work with gauge systems and using gauge invariant functions we study its quantum counterpart and we find if all these operators are self adjoint or not. Our study is divided in two cases, when we choose clock or clocks that its Poisson brackets with the set of constraints is one or it is different to one. We show some transition amplitudes.1. IntroductionThe study of gauge systems or systems with first class constraints is an outstanding branch of theoreticalphysics and mathematics, its importance lies in the fact that a major number of physical systems have firstclass constraints, including parametric systems, quantum electrodynamics, the standard model, generalrelativity and a lot of systems with a finite number of degrees of freedom and so on. A special caseof these theories are the covariant systems. In this case the canonical Hamiltonian vanishes and thesystem is invariant under the reparametrization of the coordinates. In consequence the time is not a prioridefined. Let us consider a phase space with coordinates (q1 , . . . , qn , p1 , . . . , pn ) and first class constraints γa ,where the set of constraints obeys, {γa , γb } = Cab c γc , (1)and a = 1, . . . , m and m is the number of first class constraints. We take the action principle, σ2 S[q, p, λa ] = pi q i − λa γa dσ, ˙ (2) σ1then if we vary the action we can obtain the equations of motion for our system. Now, we define agauge invariant function R or a complete observable as a phase space function such that the Poissonbrackets with the full set of constraints is zero, i. e. {R, γa } = 0. Then, all the phase space functionssuch that its Poisson bracket with all the constraints is not zero are called partial observables. If we takem partial observables as clocks T1 , . . . , Tm , and a partial observable f , then we can find a completeobservable F or a gauge invariant function (see [1] and [2] for instance). Now, our aim is to considerPublished under licence by IOP Publishing Ltd 1
  3. 3. XIV Mexican School on Particles and Fields IOP PublishingJournal of Physics: Conference Series 287 (2011) 012043 doi:10.1088/1742-6596/287/1/012043what happen to the quantum level. In that case we study two problems, the first problem is to define ˆour set of clocks all along the real line and the second is the self adjoint character for the F operator,where F is a complete observable. If Dim[Ker(F ˆ + ı)] = Dim[Ker(F − ı)], then F is self adjoint; if ˆ ˆ ˆ ˆ ˆDim[Ker(F + ı)] = Dim[Ker(F − ı)], then F is not self adjoint and it has not self adjoint extensions(see [3] for instance). In the covariant systems we have two problems first for arbitrary defined clock the ({Ti , Tj }) ({Ti , γj }) Det , (3) ({γi , Tj }) ({Di , Dj })is zero for some regions at the phase space and then clocks are not globally well defined, the second ˆproblem is that F is not self adjoint, if we find one or two of the previous problems we propose that theclocks must be selected in such a way that {Ti , γj } = δij (see [4] for instance) and we show in this workthat this form of selection for the time solve the problems for several systems.2. Non relativistic parametric free particleIn this case, the constraint for the one dimensional non-relativistic free particle is, p2 D = p0 + = 0, (4) 2mand the phase space coordinates are (x0 , x, p, p0 ).2.1. Example 1For the first example, we can take the T clock like, T = x0 − ax, (5)and the partial observable like f = x, if we make that we will obtain the complete observable: p q + mτ px0 F = p , q =x− . (6) 1 − am mHowever, we find a first problem for the present system, in this case we have {T, T } {T, D} 0 1 − ap m Det = ap {D, T } {D, D} m −1 0 2 ap = 1− , (7) mand our choice for the time is not defined all along the real line. ˆ ˆ ˆ Now, we consider the pair of equations (F + ı)ψ+ = 0 and (F − ı)ψ− = 0 to determine if F is a selfadjoint operator or it is not. In this case the solutions are: ap ıp2 τ ap2 ψ± = r± 1 − exp p− , (8) m 2m 2mwhere r+ and r− are constants, then, ap ap2 |ψ± |2 = |r± |2 |1 − | exp 2p − ; (9) m m ˆψ+ is not square integrable and ψ− is square integrable, then Dim[Ker(F + ı)] = 0, Dim[Ker(F + ˆ ˆı)] = 1 and we have found a second problem: the F operator is not self adjoint (see [3] for instance). 2
  4. 4. XIV Mexican School on Particles and Fields IOP PublishingJournal of Physics: Conference Series 287 (2011) 012043 doi:10.1088/1742-6596/287/1/0120432.2. Example 2We will study our previous system and we correct the problems of the previous example, we will use Tsuch that {T, D}=1 (see [4] for instance) and the general solution for T will be: mx mx T = + f p, x0 − , p0 , (10) p pthen {T, T } {T, D} 0 1 Det = = 1, {D, T } {D, D} −1 0and our T clock is well defined all along the real line. To simplify we consider the clock, mx mx T = + αp + β x0 − + ρp0 , (11) p pwith {T, D} = 1 and for the partial observable f = x we obtain, pτ αp2 βpx0 ρp0 p F = − + βx − − . (12) m m m m ˆ ˆ ˆTo determine the self adjointness for F we must find Dim[Ker(F + ı)], Dim[Ker(F − ı)] and compare ˆ ˆour results, with the previous purpose we consider (F + ı)ψ+ = 0 and (F − ı)ψ− = 0. We take, p2 ψ± (p, p0 ) = δ p0 + , (13) 2m ˆand the solutions for (F ± ı)ψ± = 0 are: p p2 τ αp3 ρp4 g± = r± exp +ı −ı +ı 2 β 2mβ 3mβ 8m β p2 ρ(2mp0 + p2 ) − ı , (14) 4m2 βin this case we define, p2 g (p) = g p, − ˜ (15) 2mand, ∞ ψ1 |ψ2 F is = g∗ g dp˜1 (p)˜2 (p), (16) −∞ˆF is self adjoint and we have not the second problem of the first example. ˆ Now, we take the eigenvalue equation Xψ = x1 ψ, then we take: p2 ψ(p, p0 ) = δ p0 + g(p) (17) 2mand, ıρp4 ıαp3 ıp2 τ ıx1 p g(p) = r1 exp − + − , (18) 8βm2 3βm 2mβ βusing the results above we obtain: 2mπβ ım(x1 − x1 )2 ψx1 ,τ |ψx,τ = |r1 |2 exp . (19) ı(τ − τ ) 2β(τ − τ )and in consequence we obtain the correct amplitude for the free particle (see [3] for instance). 3
  5. 5. XIV Mexican School on Particles and Fields IOP PublishingJournal of Physics: Conference Series 287 (2011) 012043 doi:10.1088/1742-6596/287/1/0120433. Two constraintsNow, we will extend our results for covariant systems with more constraints, 1 D1 = [−(p1 )2 + (p2 )2 + (p3 )2 ], 2 1 D2 = − [q1 p1 + q2 p2 + q3 p3 ], (20) 2the phase space coordinates are (q1 , q2 , q3 , p1 , p2 , p3 ) and, {D1 , D2 } = D1 . (21)We take similar restrictions like our previous example, in this way we define clocks all along the real lineand the quantum operator associated with a complete observable is self adjoint, with the restrictions: {T1 , D1 } = 1, {T2 , D2 } = 1, {T2 , D1 } = 0, (22)we have,   {T1 , T1 } {T1 , T2 } {T1 , D1 } {T1 , D2 }  {T2 , T1 } {T2 , T2 } {T2 , D1 } {T2 , D2 }  Det    {D1 , T1 } {D1 , T2 } {D1 , D1 } {D1 , D2 }  = 1, {D2 , T1 } {D2 , T2 } {D2 , D1 } {D2 , D2 }and the general solutions for T1 and T2 are: q1 q1 p2 q1 p3 T1 = − + f p1 , p2 , q2 + , p3 , q3 + , p1 p1 p1 p2 p1 p3 T2 = 2 ln(p1 ) + g , (p1 q2 + q1 p2 ), , p1 p2 p1 p1 (−p3 q2 + p2 q3 ) , (23) p2and our clocks are defined all along the real line.3.1. Example 1We can take as clocks, q1 T1 = − , T2 = 2 ln(p1 ), (24) p1and f = q2 as partial observable, then the complete observable will be: τ2 p2 τ2 F = (q1 p2 + q2 p1 ) exp − + τ2 exp , (25) 2 p1 2 ˆwe consider F ± ı ψ± = 0 and we obtain the solutions: τ2 ı exp(τ2 )τ1 exp( ) p2 − p2 g (p ), ψ± = p1 (p1 + p2 ) 2 2 1 ± 3 (26)then we have, 2 exp( τ2 2 |ψ± |2 = |p1 + p2 | 2 ) p2 − p2 |g± (p3 )|2 , (27) 2 1 ˆand we can choose g± (p3 ) such that ψ+ and ψ− are square integrable; Dim[Ker(F + ı)] = 1,Dim[Ker(F ˆ − ı)] = 1 and F will be self adjoint. ˆ 4
  6. 6. XIV Mexican School on Particles and Fields IOP PublishingJournal of Physics: Conference Series 287 (2011) 012043 doi:10.1088/1742-6596/287/1/0120433.2. Example 2We use now, q1 T1 = − , T2 = ln(p2 ), (28) p1as clocks and f = q1 like partial observable, then the complete observable will be: p1 τ2 F =− τ1 exp , (29) p2 2 ˆ ˆwe must solve the equations F ± ı ψ± = 0 to determine if F is self adjoint or it is not, we find thesolutions: q1 p2 τ2 ψ± = exp exp − g± (p3 ), (30) τ1 2and later we obtain, 2q1 p2 τ2 |ψ± |2 = exp exp − |g± (p3 )|2 . (31) τ1 2 ˆ ˆ ˆWe can choose g± (p3 ) such that Dim(Ker(F +ı)) = Dim(Ker(F −ı)) = 0 and F will be self adjoint.4. Conclusions and perspectivesTo conclude our work we make the following discussion. In the first example, for the one dimensionalnon-relativistic free particle we found two problems: the first problem was that, {T, T } {T, D} Det = 0, {D, T } {D, D}for a selection for T and our clock was not defined all along the real line, the second problem was thatthe operator associated with our complete observable F was not self adjoint, we correct both problemswhen we choose our clock T like a phase space function such that the Poisson bracket with the constraintis one {T, D} = 1. Now, in this case, {T, T } {T, D} Det = 1, {D, T } {D, D}and our clock is well defined all along the real line and our operator associated with our completeobservable F is self adjoint. For the case of two constraints, we begin with clocks that are conjugate to the set of constraints andwe did not find problems with the operators associated with the complete observable. For the model with two constraints we studied some partial observables with different selections forthe clocks in such a way that, ({Ti , Tj }) ({Ti , Dj }) Det = 1, ({Di , Tj }) ({Di , Dj }) ˆand we found that F is self adjoint, where F is a complete observable constructed for a partial observablef and the clocks are globally well defined. However, with a different choice for the previous determinant,we could have the problem that our clocks are not defined all along the real line and the problem to theself adjointness for the operator associated with a complete observable could be present. The following general study can be done: if you have a phase space with coordinates(q1 , . . . , qn , p1 , . . . , pn ) and first class constraints γa , in such a way that, {γa , γb } = Cab c γc , (32) 5
  7. 7. XIV Mexican School on Particles and Fields IOP PublishingJournal of Physics: Conference Series 287 (2011) 012043 doi:10.1088/1742-6596/287/1/012043where a = 1, . . . , m and m is the number of first class constraints. Then, we can choose m cloks in sucha way that {Ti , γj } = δij and a first problem is to determine if, ({Ti , Tj }) ({Ti , γj }) Det = 0, (33) ({γi , Tj }) ({Di , Dj })for all the values of Ti , if you have that, the clocks are defined all along the real line. ˆ A second general problem with our previous conditions is to determine if the operator F associatedwith the complete observable F is self adjoint, where F is obtained of a partial observable f . If ourmethod fails in some cases, we must determine these reasons in future studies. The definition of clocks and self adjoint operators associated with complete observables or gaugefunctions is open for complex systems, including general relativity.References[1] B. Dittrich, Partial and complete observables for Hamiltonian constrained systems, Gen. Rel. Grav. 39 (2007) 1891-1927.[2] B. Dittrich and T. Thiemann, Are the spectra of geometrical operators in loop quantum gravity really discrete?, J. Math. Phys. 50, 012503 (2009).[3] R. Gambini and R. A. Porto, Relational time in generally covariant quantum systems: Four Models, Phys. Rev D, 63, 105014, (2001).[4] F. Calogero and F. Leyvraz, General technique to produce isochronous Hamiltonians, J. Phys. A: Math. Theor. 40, (2007), 12931-12944. 6

×