If we had started in Eq. This coincides with the path integral formulation of classical mechanics postulated by Gozzi et al. The same representation can be derived from the classical limit of a closed-time path integral for the transition probabilities of a quantum particle in a heat bath [15,18], The path integral 28 has an interesting mathematical structure.
We may rewrite it as. By representing the delta functional in the usual way as a functional Fourier integral. It is simple to check  that S does not change under the symmetry transformations. On the formal side, and c may be identified with Jacobi fields [18,19]. By analogy with the stochastic quantization the path integral 30 can be rewritten in a compact form with the help of a superfield [17,21,15].
In terms of superspace variables we see that. To obtain the last line we Taylor expanded L and used the standard integration rules for Grassmann variables. Together with the identity we may therefore express the classical partition functions 28 and 29 as a supersymmetric path integral with fully fluctuating paths in superspace.
Let us finally add that under rather general assumptions it is possible to prove  that 't Hooft's deterministic systems are the only systems with the peculiar property that their full quantum properties are classical in the Gozzi et al. Among others, the latter also indicates that the Koopman-von Neumann operator formulation of classical mechanics  when applied to 't Hooft systems must agree with their canonically quantized counterparts. As observed in Section 2. To deal with this situation we now employ 't Hooft's procedure of Section 2.
We assume that the system 8 has n conserved irreducible charges C i , i. For convenience take the following choice. The lower bound is reached by choosing a i t C i to be non-negative.
We shall select a combination of C i which is p -independent [this condition may not necessarily be achievable for general f a q ]. In the Dirac-Bergmann quantization approach used in our previous paper , the information loss condition 12 was a first-class primary constraint. The time evolution of observable quantities, however, should not be affected by the arbitrariness of Lagrange multipliers.
This book has been written twice. After having written and published it in German in , I started allover again and rewrote the whole story for an English. quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires.
To remove this superfluous freedom one must choose a gauge. For details of this more complicated procedure see . In the Faddeev-Jackiw approach, Dirac's elaborate classification of constraints to first or second class, primary or secondary is avoided. It is therefore worthwhile to rephrase the entire development of Ref.
Path integration via summation of perturbation expansions and applications to totally reflecting boundaries, and potential steps - Grosche, Christian Phys. Fourier Decomposition. Magnetic Fields. ST Phys. Quan Phys. Gozzi, J.
The information loss condition may now be introduced by simply adding to the Lagrangian 18 a term enforcing. Apart from an irrelevant total derivative, this changes the derivative term to , with. Eliminating x 1 also in the Hamiltonian H we obtain a reduced Hamiltonian H R x , so that we are left with a reduced Lagrangian.
At this point one must worry about the notorious operator-ordering problem, not knowing in which temporal order and must be taken in the kinetic term. A path integral in which the kinetic term is coordinate-dependent can in general only be defined perturbatively, in which all anharmonic terms are treated as interactions. The partition function is expanded in powers of expectation values of products of these interactions which, in turn, are expanded into integrals over all Wick contractions, the Feynman integrals.
Each contraction represents a Green function. Thus, the Feynman integrals run over products of distributions and are mathematically undefined. Fortunately, a unique definition has recently been found. It is enforced by the necessary physical requirement that path integrals must be invariant under coordinate transformations . The Lagrangian is processed further with the help of Darboux's theorem . This allows us to perform a non-canonical transformation x i z s , z r which brings L R to the canonical form. Darboux's theorem ensures that such a transformation exists at least locally.
The variables z r are related to zero modes of the matrix f ij which makes it non-invertible. Each zero mode corresponds to a constraint of the system.
In Dirac's language these would correspond to the secondary constraints. In general, z , z is a nonlinear function of. One now solves as many as possible in terms of remaining z 's, which we label by , i.
If z , z happens to be linear in , we obtain the constraints. We now repeat the elimination procedure until there are no more z -variables.
The surviving variables represent the true physical degrees of freedom. Use of the path integral may now proceed along the same lines as in Ref. In fact, when Darboux's transformation is global it is possible to show  that the resultant path integral representation coincides with the one in .
In this paper we have presented a path-integral formulation of 't Hooft's quantization procedure, in the line of what has been done recently in Ref. With respect to our previous work, we have here utilized the Faddeev-Jackiw treatment of singular Lagrangians  which present several advantages with respect to the usual Dirac-Bergmann method for constrained systems.
In particular, one does not require the Dirac-Bergmann distinction of first and second class, primary and secondary constraints used in . The Faddeev-Jackiw method is also convenient in imposing 't Hooft's information loss condition. Although it appears that the Faddeev-Jackiw method allows for considerable formal simplifications of the treatment, more analysis is needed in order to compare with our previous results of Ref. This is object of work in progress . Note finally that according to analysis in Section 5, when we start with the N -dimensional classical system q variables then the emergent quantum dynamics has N -1 dimensions z variables.
This reduction of dimensionality reflects the information loss. Our result supports the strong version of the holographic principle , namely that the deterministic degrees of freedom of a system scale with the bulk , while the emergent quantum degrees of freedom i. The authors acknowledge an instigating communication with R. Jackiw and very helpful discussions with E. Gozzi, J. Pons, and F. Elze, Lecture Notes in Physics, Vol. Susskind, L. Thorlacius, and J. Uglom, Phys. D 48 , Bousso, Rev. Blasone, P. Jizba and H. Kleinert, accepted to Phys. Jizba and G. Vitiello, J. Elze, Phys.
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