Multiple Groenewold Products: from path integrals to semiclassical correlations

Transcription

1 Multiple Groeewold Products: from path itegrals to semiclassical correlatios

2 1. Traslatio ad reflectio bases for operators Traslatio operators, correspod to classical traslatios, withi the classical phase space, They form a complete operator basis, so that ay operator with the expasio coefficiets: This is the chord symbol of the operator Â.

3 The Fourier trasform of the traslatio operators defies uitary reflectio operators, correspodig to the classical reflectios,. A arbitrary operator, Â, ca be decomposed i this basis: such that the expasio coefficiet is the Weyl-Wiger symbol for Â. I the case of the desity operator, W(x) is just the Wiger fuctio. Grossma Royer

4 The chord symbol for a product of operators, Aˆ Aˆ... ˆ, ~ ~ ~ 1 A A... A 1 1 ~ d... d A... A tr T T... T N 1 is determied by the quatum traslatio group: Tˆ Tˆ ˆ ˆ i... T T exp D ~ ˆ,...,. The fial traslatio operator correspods to the overall classical traslatio ad D,..., 1 1 is the symplectic area of the (+1)-sided polygo formed by the traslatios. ˆ A 1 ˆ

5 Tecelatig the polygo with triagles, specifies D,..., [... (... ) ]. 1 1 This area reflects the associative, but ocommutative properties of the operator product. 5 3 Projectios of phase space polygos oto cojugate plaes may have complex self-itersectios: 1 4 2

6 ... A (x), 1 The Weyl symbol of this product, the Fourier trasform of the chord symbol, is eatly writte i terms of the multivariable fuctio, ' A... A 1 '(x,...,x ) 1 A d... d1 i ~ i exp 1,..., ( )exp, 2 N 1 D Aj j x j j j such that the Weyl represetatio of the product is just ' A 1... A '(x,...,x) A... A (x). 1

7 Except for the polygoal factor, would be just a product of Weyl symbols, A ( x )... A 1 (x ). Thus, the full multiple Fourier trasform is simply expressed 1 as ' A... A 1 '(x,...,x ) 1 ' A... A 1 '(x,...,x ) 1 exp i D,..., A (x )... A (x 1 x1 x 1 We have a multiple Groeewold-Moyal-star product. ). For a sigle pair of operators, we regai A A ( x) exp 2 1 i (x ) (x ) 1 1 x 1 x 2 2 x 2 x 2 x1 x A A 2

8 What about direct use of the Weyl represetatio? The simplest case is for a eve umber of operators: dx... dx A... A (x) 1 A (x )... A (x ) tr Rˆ Rˆ... Rˆ 1 N 1 1 x x ( ) The properties of the quatum affie group (reflectio x reflectio = traslatio) (reflectio x traslatio = reflectio) the lead to: dx... x... (x) 1 (x )... (x ) exp 1(x,x1,...,x ). 1 d i A A A A N 1 1 ( ) x 1. Agai we have a polygo determiig the phase, but 1 ( x,x,...,x ) 1 is ow specified by the cetres of its sides. x 1

9 2. Path itegral for the Weyl propagator: semiclassical limit The Weyl Hamiltoia, H(x), is close to the classical Hamiltoia, withi order of. I the limit of small times, the Weyl propagator, i.e., the Weyl symbol for the evolutio operator,, Uˆt is U t (x) exp (x). 0 i t H t The the path itegral for fiite times is merely the product formula, itself: U t lim x... x ( x) 1 i t d d exp 1(x,x1,...,x ) ( '). ( ) H x N ' 1

10 If the Hamiltoia separates ito kietic ad potetial eergies, this is the Weyl trasform of the Feyma path itegral: dp q q- i - t ). q Uˆ q U (p, ) exp p (q - q- (2 ) 2 t N The phase added i the trasform fills i the area betwee the polygo ad the q-axis. The full area ca the be covered by thi strips. 5

11 Statioary phase evaluatio, for each cetre 1 t H J j x J x j I short, the trajectory at each cetre must be taget to the respective side of the polygo. j t x j. x j, demads that: I the limit,, the statioary polygo defies a sigle classical trajectory, with its edpoits cetred at x. The semiclassical Weyl propagator is the x 1 if there is a sigle trajectory with cetre x Berry

12 The cetre actio or cetre geeratig fuctio, S(x), specifies a fiite evolutio through Hamilto s equatios : The full cetre actio is S(x)= s(x) Et. s(x) Usual geeratig fuctios specify a trajectory by a pair of positios, (q, q ) while mometa (p, p ) are free. Here, we have a fixed cetre with free chord. The moodromy matrix, M, determies the liearized trasformatio, betwee the tips of the classical trajectory.

13 3. Compoud uitary operators Multiple evolvig correlatios amog observables: where each of the operators udergoes a Heiseberg evolutio: Defie the itermediate steps: the icludig Loschmidt echo, or fidelity.

14 Thus, the evolvig correlatio becomes where the kerel for the evolutio for the iitial correlatio is defied as But the reflectio operators are also uitary, so that this sequece ca be cosidered as a sigle compoud uitary operator. It defies a quatum evolutio correspodig to a classical compoud caoical trasformatio.

15 Assume that the compoud Weyl propagator, U(x), shares the stadard semiclassical form as each idividualpropagator, U ( x '). That is, assume that the compoud classical actio is

16 Likewise, the amplitude of the compoud propagator is determied by the moodromy matrix of the full motio: The product of the liearised trasformatio for each segmet: reflectios evolutios The moodromy matrices for the reflectios are idepedet of the positio of their reflectio cetres, but M ad S(x) deped o all the cetres, that parametrize a family of caoical trasformatios.

17 Now oe requires tr Û, but the trace of ay operator equals the phase space itegral of its Weyl represetatio: The oly explicit depedece of U(x) o x lies i Sice the chord cetred o x depeds oly o the other cetres, the

18 But if the chord cetred o x is zero, the selected trajectory is periodic! The statioary phase evaluatio:

19 4. Iitial value represetatio The appropriate trajectories for a semiclassical propagator are determied by boudary coditios. If the fiite evoutio is specified by Hamilto s differetial equatios, there is a practical root search problem to fid the trajectory. Also, i the case of the trace, oe must search for the periodic trajectories. A further problem cocers caustics:

20 There may be several trajectories with the same cetre: A pair of chords coalesces for a cetre, x, o a cetre caustic. The caustic sigularities of the Weyl propagator are loci of cetres, at which a eigevalue of M, 1. I the case of the trace of the propagator, the caustics arise at periodic orbit bifurcatios: They occur alog codimesio-1 surfaces i the parameter space

21 Let us the reiterpret tr Û as the Weyl represetatio of a reduced compoud uitary operator: that is, This has oe less reflectio tha Û, but it has a aalogous semiclassical approximatio:

22 The same figure as before is ow iterpreted as _ a ope polygoal lie, goig from to x. x 0 0 Each brach of the geeratig fuctio S x 0 is costructed _ from a compoud trajectory that satisfies x 0 x 0 2 x0. But there is still a root search ad caustics

23 The iitial value represetatio (IVR) ow results from the chage of variable i the itegral for the correlatio Thus, oe chages the itegratio variable to the iitial value of the classical trajectory: The Jacobia is det x x 0-0

24 Thus, the semiclassical approximatio for the evolvig correlatio becomes: where ow all classical variables are determied by the iitial value of the compoud trajectory:. x _ 0 No more root search ad o more caustics!

25 5. A example: IVR for the quatum fidelity The Loschmidt echo for differet forward ad back evolutios, L( t) i i exp Hˆ t exp Hˆ t ca be expressed i terms of a echo operator, a simple compoud operator, with Iˆ Tˆ 0, istead of R : ˆx

26 Thus, we obtai the IVR: This is exact for a pair of harmoic oscilators.

27 Vaiceck s dephasig represetatio results by approximatig the actio withi classical perturbatio theory as the time itegral of H H ( x) H ( x) alog a sigle trajectory:

28 5. Discussio: The exchage of focus from the idividual semiclassical propagator to complete evolvig correlatios pays off! Some care eeds still to be take: i. Geeral rules for phase evaluatio through caustics: These become zeroes of the itegrad, leadig to sig chages: i. Numerical computatios for oliear evolutios (Compariso with Herma-Kluk computatios). iii. Adaptatio to ouitary (Markovia) evolutio. iv. Semiclassical evaluatio for reduced desity operators.

29 Refereces: - Phase space path itegral for the Weyl propagator, OA 1992, Proc. R. Soc.A 439, The Weyl Represetatio i classical ad quatum mechaics OA 1998, Phys. Reports 295, The Weyl represetatio o the torus Rivas AMF ad OA 1999, A. Phys. NY 276, Etaglemet i phase space OA 2009, Lecture Notes i Physics 768, Semiclassical evolutio of dissipative markovia systems OA, Rios PM ad Brodier O 2009, J. Phys. A (29p) -Metaplectic sheets ad caustic traversals i the Weyl represetatio OA ad Igold G-L 2014, J. Phys. A 47, Semiclassical evoluio of correlatios betwee observables OA ad Brodier O 2016, J. Phys. A 49, (19pp) - Represetatio of superoperators i double phase space Saraceo M ad OA 2016, J. Phys. A 49, (23pp)