Stochastic Quantization Method with Field-Dependent Metric

From constrained stochastic processes to the nonlinear sigma model. Two old problems revisited

“…For comparison, the standard treatment of constrained path integrals, see for instance [36], produces complicated jacobian determinants of block matrices. The path integral expressions obtained in this way are very useful at a theoretical level, because they allow for example to check if the Parisi-Wu quantization scheme delivers in the equilibrium limit the desired generating functional of the theory to be quantized [8,9], but are very cumbersome in concrete calculations of physical observables. The method discussed in this work produces instead relatively simple generating functionals.…”

“…Stochastic systems in the presence of constraints are studied in connection with several physical problems [1][2][3][4][5][6][7][8][9]. In the absence of constraints, a very useful method in order to solve stochastic differential equations like those of Fokker-Planck consists in expressing their solution in path integral form [10][11][12][13].…”

“…The derivation of the path integral solutions of the Fokker-Planck equations becomes however complicated when constraints are added. Various approaches have been proposed in order to cope with constrained stochastic differential equations [8,9,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], but the available path integral formulations for statistical systems out of equilibrium still remain awkward to be used in practical applications.…”

“…A considerable advance with respect to [31] is the inclusion in the treatment of the external forces, that were previously missing. While with the usual approaches the path integral of the generating functional contains extra degrees of freedom (the Lagrange multipliers) [8,9,34,35] or, alternatively, is complicated by functional determinants of block matrices [8,9,36], in the present approach the functional determinants are relatively simple and can be computed analytically.…”

The Stochastic Quantization Method in Phase Space and a New Gauge Fixing Procedure

“…[5] Moreover we can observe it induces the Lee-Yang term [10] in equilibrium Fokker-Planck distribution. [11] The Fokker-Planck equation in this case is…”