variable. This is the essence of stochastic quantization [60–62], which is thus recognized as a schematic imple- mentation of micro-canonical quantization [63]. 7 denote the dynamical variables (fields) collectively by A and the θ -energy associated with a particular set of these variables as H [ A ]. Observed over long times θ , the sys- tem will visit all points on the hypersurface defined by δ ( H [ A ] − H$_{0}$ ), where H$_{0}$ is the initial θ -energy of our sys- tem. The long θ -time average of an observable O [ A ] is then equal to the microcanonical average: Of course, the micro-canonical average can be gener- ated by any other appropriate method, e. g. by a de- terministic, but ergodic process in an additional physical (not “auxiliary”) time dimension. Such a process is nat- urally realized when the evolution of the five-dimensional system in the additional time variable θ is strongly chaotic. The method of defining the vacuum expectation value of a system by means of the ergodic average over a chaotic trajectory has been called chaotic quantization [47, 64]. where V ( H$_{0}$ ) is the volume of the constant- H hypersur- face. The last expression in (22) is the definition of the quantum mechanical average of the observable O in a formalism called micro-canonical quantization [52, 53]. Using the Fourier representation of the delta function, the micro-canonical average can be written in the form An example of a physical system that thus generates its own quantum analogue in a reduced dimension is the classical lattice gauge theory [48]. A numerical study comparing the “self-quantizing” five-dimensional classi- cal gauge theory with the canonically quantized gauge theory in four dimensions, which was undertaken by Bir´ o and M¨ uller for the compact U(1) lattice gauge the- ory [65], showed excellent agreement for the expectation value of the Polyakov loop as a function of the gauge cou- pling if Planck’s constant ℏ is identified with the product T$_{5}$a , where a is the lattice spacing. For systems with an infinite number of degrees of free- dom, such as field theories, the microcanonical average is known to agree with the canonical average under very general conditions [52]. When this equivalence holds, we can express the expectation value of the observable O as In order to describe physics in real physical time, one must analytically continue the euclidean time coordinate τ to the real time coordinate t = − iτ . For the canoni- cal average (24), this is exactly what is usually done in quantum field theory, where the Minkowski space vacuum expectation value of an observable O is defined by the analytic continuation of the euclidean space functional integral. The critical question is whether the same an- alytic continuation also applies to, and makes sense for, the micro-canonical and ergodic time averages (22). The analytic continuation involves a number of sub- tleties. First, the real-time analogue of the functional in- tegral (24) does not involve a real integrand and is thus not mathematically well defined. For the micro-canonical average (22) the problem is that the analytically contin- ued θ -energy (the Minkowski space action) is not posi- tive definite, and thus the constant- H hypersurface does not have a finite volume, even for a field theory with in- frared and ultraviolet cut-offs. In the stochastic average, the mathematical subtleties reside in the definition of a Langevin process for a system governed by a Minkowski space action with indefinite sign. There has been a limited amount of work on stochas- tic quantization in Minkowski space [66, 67]. For cer- tain systems, a complex Langevin process can be defined and simulated, which generates the appropriate micro- canonical ensemble [68, 69]. In other cases, the com- plex Langevin process can be stabilized by a suitable ker- nel [70]. Recently, complex Langevin evolution has been applied to generate the real-time quantum evolution of The micro-canonical ensemble has been widely used in lattice gauge theory to calculate vacuum expectation val- ues of various operators [54–59]. In practice, the micro- canonical ensemble is generated by a stochastic trajec- tory obeying a Langevin equation in an auxiliary “time” Upon the identification β = ℏ − $^{1}$, the expression (24) be- comes the usual definition of the (canonical) quantum mechanical expectation value of the observable O as a functional integral in euclidean space. H [ A ] is recognized as the euclidean space action associated with the system. In our five-dimensional space-time H is simply the energy variable associated with the additional time dimension. The quantum of action (Planck’s constant) ℏ appears as the quasi-temperature associated with the ergodic finite- energy trajectory of the system in the five-dimensional space-time. Note that this definition of the expectation variable does not involves a thermal ensemble, only a ( θ )- time average over an ergodic, classical trajectory in the five-dimensional space-time. where Z ( β ) = ∫ D A e − βH [ A ] is a partition function and the parameter β is, as usual, defined by the condition 〈O〉 = Z ( β ) − 1 ∫ D A e − βH [ A ] O [ A ] , (24) β ∫ D A e − βH [ A $^{]}$H [ A ] ≡ − β ∂Z ∂β = βH$_{0}$. (25) B. Stochastic and chaotic quantization C. Analytic continuation 〈O〉 = V ( H$_{0}$ ) − 1 ∫ ∞ −∞ d ˜ θ 2 π ∫ D A e i ( H [ A ] − H$_{0}$ ) ˜ θ O [ A ] . (23) 〈O〉 = lim θ →∞ 1 θ ∫ θ 0 dθ ′ O [ A ( θ $^{′}$)] = V ( H$_{0}$ ) − 1 ∫ D A δ ( H [ A ] − H$_{0}$ ) O [ A ] , (22)