• Share
  • Email
  • Embed
  • Like
  • Save
  • Private Content

Loading…

Flash Player 9 (or above) is needed to view presentations.
We have detected that you do not have it on your computer. To install it, go here.

Like this document? Why not share!

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

on

  • 204 views

Poster for the XIV Mexican School on Particles and Fields; 4 november 2010- 12 november 2010; Morelia , Michoacán.

Poster for the XIV Mexican School on Particles and Fields; 4 november 2010- 12 november 2010; Morelia , Michoacán.

Statistics

Views

Total Views
204
Views on SlideShare
201
Embed Views
3

Actions

Likes
0
Downloads
0
Comments
0

1 Embed 3

http://www.linkedin.com 3

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    Poster Gauge systems and functions, hermitian operators and clocks as conjugate functions for the constraints Poster Gauge systems and functions, hermitian operators and clocks as conjugate functions for the constraints Document Transcript

    • Gauge systems and functions, hermitian operators and clocks as conjugate functions for the constraints Vladimir Cuesta 1,† , Jos´ David Vergara e 1,†† and Merced Montesinos 2,∗ 1 Instituto de Ciencias Nucleares, Universidad Nacional Aut´noma de M´xico, 70-543, Ciudad de M´xico, M´xico o e e e 2 Departamento de F´ ısica, Centro de Investigaci´n y de Estudios Avanzados del Instituto Polit´cnico Nacional, Instituto Polit´cnico Nacional 2508, San Pedro o e e Zacatenco, 07360, Gustavo A. Madero, Ciudad de M´xico, M´xico e e p2 Abstract in this case we define g (p) = g p, − ˜ 2m ∞ g∗ ˆ and ψ1 |ψ2 F is = −∞ dp˜1 (p)˜2 (p), F is self adjoint g We work with gauge systems and using gauge invariant functions we study its quantum and we have not the second problem of the first example. counterpart and we find if all these operators are self adjoint or not. Our study is divided in two ˆ p2 Now, we take the eigenvalue equation Xψ = x1 ψ, then we take ψ(p, p0 ) = δ p0 + g(p) cases, when we choose clock or clocks that its Poisson brackets with the set of constraints is one 2m or it is different to one. We show some transition amplitudes. ıρp4 ıαp3 ıp2 τ ıx1 p and g(p) = r1 exp − + − , with the previous result we obtain: 8βm2 3βm 2mβ β Introduction   The study of gauge systems or systems with first class constraints is an outstanding branch 2 2mπβ ım(x1 − x1 )2 of theoretical physics and mathematics, its importance lies in the fact that a major number ψ |ψx,τ = |r1 | exp  . (5) x1 ,τ ı(τ − τ ) 2β(τ − τ ) of physical systems have first class constraints, including parametric systems, quantum electro- dynamics, the standard model, general relativity and a lot of systems with a finite number of degrees of freedom and so on. A special case of these theories are the covariant systems. In this This is the correct result. case the canonical Hamiltonian vanishes and the system is invariant under the reparametrization of the coordinates. In consequence the time is not a priori defined. Two constraints Let us consider a phase space with coordinates (q1 , . . . , qn , p1 , . . . , pn ) and first class The constraints for our model are 1 D1 = 1 [−(p1 )2 + (p2 )2 + (p3 )2 ], D2 = − 1 [q1 p1 + constraints γa , where the set of constraints obeys {γa , γb } = Cab c γc and a = 1, . . . , m 2 2 q2 p2 +q3 p3 ], and {D1 , D2 } = D1 . We take similar restrictions like our previous example, in this and m is the number of first class constraints. We take the action principle S[q, p, λa ] = way we define clocks all along the real line and the quantum operator associated with a complete σ2 ˙i a σ1 pi q − λ γa dσ, then if we vary the action we can obtain the equations of motion for our observable is self adjoint, with the restrictions {T1 , D1 } = 1, {T2 , D2 } = 1 and {T2 , D1 } = 0, system. Now, we define a gauge invariant function R or a complete observable as a phase space we have, function such that the Poisson brackets with the full set of constraints is zero, i. e. {R, γa } = 0.   Then, all the phase space functions such that its Poisson bracket with all the constraints is not {T1 , T1 } {T1 , T2 } {T1 , D1 } {T1 , D2 } zero are called partial observables. If we take m partial observables as clocks T1 , . . . , Tm , and  {T2 , T1 } {T2 , T2 } {T2 , D1 } {T2 , D2 }  Det   = 1, a partial observable f , then we can find a complete observable F or a gauge invariant func-  {D1 , T1 } {D1 , T2 } {D1 , D1 } {D1 , D2 }  tion (see [1] and [2] for instance). Now, our aim is to consider what happen to the quantum {D2 , T1 } {D2 , T2 } {D2 , D1 } {D2 , D2 } level. In that case we study two problems, the first problem is to define our set of clocks all ˆ along the real line and the second is the self adjoint character for the F operator, where F and the general solutions for T1 and T2 are: ˆ ˆ ˆ is a complete observable. If Dim[Ker(F + ı)] = Dim[Ker(F − ı)], then F is self adjoint; if Dim[Ker(F ˆ + ı)] = Dim[Ker(F − ı)], then F is not self adjoint and it has not self adjoint ex- ˆ ˆ q1 q1 p2 q1 p3 tensions (see [3] for instance). In the covariant systems we have two problems first for arbitrary T1 = − +f p1 , p2 , q2 + , p3 , q3 + , p1 p1 p1 defined clock the   p2 p1 p3 p1 {Ti , Tj } {Ti , Dj } T2 = 2 ln(p1 ) + g , (p1 q2 + q1 p2 ), , (−p3 q2 + p2 q3 ) , (6) Det   , p1 p2 p1 p2 {Di , Tj } {Di , Dj } and our clocks are defined all along the real line. is zero for some regions at the phase space and then clocks are not globally well defined, the ˆ second problem is that F is not self adjoint, if we find one or two of the previous problems we Example 1 propose that the clocks must be selected in such a way that {Ti , Dj } = δij (see [4] for instance) q We can take T1 = − 1 , T2 = 2 ln(p1 ) as clocks and f = q2 as partial observable, then the and we show in this poster that this form of select the time solves the problems for several p1 systems. complete observable will be: Non relativistic parametric free particle F = (q1 p2 + q2 p1 ) exp − τ2 + p2 τ2 exp τ2 , (7) 2 p1 2 In this case, the constraint for the one dimensional non-relativistic free particle is p2 D = p0 + = 0 and the phase space coordinates are (x0 , x, p, p0 ). we consider ˆ F ± ı ψ± = 0 and we obtain the solutions: 2m Example 1 ı exp(τ2 )τ1 exp τ2 2 2 2 ψ± = p1 (p1 + p2 ) p2 − p1 g± (p3 ), (8) For the first example, we can take the T clock like T = x0 − ax and the partial observable like f = x, if we make that we will obtain the complete observable: τ2 2 exp 2 then we have |ψ± |2 = |p1 + p2 | 2 p2 − p2 2 1 |g± (p3 )|2 and we can choose g± (p3 ) p q+τ px0 ˆ ˆ F = m , q = x− . (1) such that ψ+ and ψ− are square integrable; Dim[Ker(F + ı)] = 1, Dim[Ker(F − ı)] = 1 and p 1−a m ˆ F will be self adjoint. m However, we find a first problem for the present system, in this case we have Example 2 q 0 1− ap ap 2 We use now T1 = − 1 and T2 = ln(p2 ) as clocks and f = q1 like partial observable, then {T , T } {T , D} m p1 Det = ap = 1− , {D, T } {D, D} −1 0 m the complete observable will be: m p1 τ2 and our choice for the time is not defined all along the real line. F = − τ1 exp , (9) ˆ ˆ Now, we consider the pair of equations (F + ı)ψ+ = 0 and (F − ı)ψ− = 0 to determine if p2 2 ˆ F is a self adjoint operator or it is not. In this case the solutions are: we must solve the equations ˆ ˆ F ± ı ψ± = 0 to determine if F is self adjoint or it is not, we find    ap ıp2 τ ap2 the solutions: ψ± = r± 1− exp  p −  , (2) q1 p2 τ2 m 2m 2m ψ± = exp exp − g± (p3 ), (10) τ1 2 2q1 p2 τ ap ap2 and later we obtain |ψ± |2 = exp exp − 2 |g± (p3 )|2 . We can choose g± (p3 ) such where r+ and r− are constant, then |ψ± |2 = |r± |2 |1 − | exp 2p − ; ψ+ is not τ1 2 m m ˆ ˆ that Dim(Ker(F ˆ + ı)) = Dim(Ker(F − ı)) = 0 and F will be self adjoint. ˆ ˆ square integrable and ψ− is square integrable, then Dim[Ker(F + ı)] = 0, Dim[Ker(F + ı)] = 1 ˆ and we have found a second problem: the F operator is not self adjoint (see [3] for instance). Conclusions and perspectives Example 2 In the first example for the one dimensional non-relativistic free particle we found two problems: our clock was not defined all along the real line and the operator associated with a We will study our previous system and we correct the problems of the previous example, complete observable was not self adjoint, we correct the problem when we choose our clock like we will use T such that {T , D}=1 (see [4] for instance) and the general solution for T will be: a phase space function such that the Poisson bracket with the constraint is one. T = mx + f p, x0 − mx , p0 , then For the case of two constraints, we begin with clocks that are conjugate to the set of con- p p straints and we did not find problems with the operators associated with the complete observable. For the one dimensional non-relativistic free particle we studied the partial observable f = x {T , T } {T , D} 0 1 with different choices for the clocks and we correct some problems for the system. However, if Det = = 1, {D, T } {D, D} −1 0 we choose the partial observable f = x0 , we can choose a clock such that {T , D} = 1 and F is ˆ not self adjoint, where F is a complete observable constructed for f = x0 , we have made that and our T clock is well defined all along the real line. study, but is not presented in the present poster. We continue, for the clock T = mx + αp + β x0 − mx + ρp0 with {T , D} = 1 and the For the model with two constraints we studied some partial observables with different se- p p   partial observable f = x we obtain {Ti , Tj } {Ti , Dj } lections for the clocks in such a way that Det   = 1 and we {Di , Tj } {Di , Dj } pτ αp2 βpx0 ρp0 p F = − + βx − − . (3) ˆ found that F is self adjoint, where F is a complete observable constructed for a partial observ- m m m m able f and the clocks are globally well defined. However, with a different choice for the previous determinant, we have the problem that our clocks are not defined all along the real line and the ˆ ˆ ˆ To determine the self adjointness for F we must find Dim[Ker(F + ı)], Dim[Ker(F − ı)] and problem to the self adjointness for the operator associated with a complete observable is open. ˆ ˆ compare our answers, with the previous purpose we consider (F + ı)ψ+ = 0 and (F − ı)ψ− = 0. The self adjoint character for operators associated with complete observables in field theory is open and it must be studied. p2 ˆ We take ψ± (p, p0 ) = δ p0 + and the solutions for (F ± ı)ψ± = 0 are: The definition of clocks and self adjoint operators associated with complete observables or 2m gauge functions is open for complex systems, including general relativity.   g± = r± exp  p +ı p2 τ −ı αp3 +ı ρp4 −ı p2 ρ(2mp0 + p2 ) , (4) References β 2mβ 3mβ 8m2 β 4m2 β [1] B. Dittrich, Gen. Rel. Grav. 39 (2007) 1891-1927, [2] B. Dittrich and T. Thiemann, J. Math. Phys. 50, 012503 (2009), 1 ,† vladimir.cuesta@nucleares.unam.mx [3] R. Gambini and R. A. Porto, Phys. Rev D, 63, 105014, (2001). [4] F. Calogero and F. Leyvraz, J. Phys. A: Math. Theor. 40, (2007), 12931-12944. 1 ,†† vergara@nucleares.unam.mx 2 ,∗ merced@fis.cinvestav.mx 1 The phase space coordinates are (q , q , q , p , p , p ). 1 2 3 1 2 3