Physics Letters B 288 ( 1992 ) 129-139 North-Holland PHYSICS LETTERS B Stochastic quantization of bottomless systems based on a kerneled Langevin equation Satoshi Tanaka, Mikio Namiki, Ichiro Ohba, Masashi Mizutani, Nobuyuki Komoike and Masahiko Kanenaga Department of Physics, Waseda University, Tokyo169, Japan Received 13 January 1992; revised manuscript received 4 May 1992 We propose a new method for quantizing systems with bottomless action functionals. Our method is based on the kerneled Langevin equation of the stochastic quantization method. We prove that our kerneled Langevin equation describes a stochastic process with a thermal equilibrium state. The proof makes the relation between our method and the conventional path-integral quantization clear. We check numerically in a simple model that dynamical evolution leads to the desired equilibrium state. We also calculateconvergent nonperturbative expectation values that are consistent with the perturbative results in the case of a small coupling constant. We also confirm that our method gives the same perturbative results as the conventional path-integral quanti- zation method. Finally, to augment understanding, we discuss a (d+ 1)-dimensional path-integral representation of our method. I. Introduction Since the stochastic quantization method (SQM) was first proposed [ 1 ], many authors have pointed out that this method extends the territory of quantum theory. For example, Greensite and Halpern proposed a scheme to apply the SQM to bottomless systems [2]. Their scheme is based on the (d+ 1 )-dimensional path-integral representation of the SQM [3-6], and it was recently applied to two-dimensional gravity [7]. This scheme is interesting but unsatisfactory, in that the relation to the conventional d-dimensional path-integral quantization method is unclear. The key problem is that the stochastic process corresponding to the (d+ 1 )-dimensional path-integral representation does not have an equilibrium state, reflecting the unboundedness of the action functional, and so the Feynman path-integral measure is not given as an equilibrium Fokker-Planck distribution. Here we propose a new stochastic quantization scheme to quantize bottomless systems based on a kerneled Langevin equation [8] ~1. Our kerneled Langevin equation describes a stochastic process which has an equilib- rium state, and the equilibrium Fokker-Planck distribution makes the relation between our method and the conventional d-dimensional path-integral quantization method clear. The presence of an equilibrium state makes it straightforward to apply our scheme to two-dimensional gravity, for example. Let us briefly review the SQM based on a kerneled Langevin equation. A scalar field O characterized by an action functional S[¢] is quantized by setting up the following kerneled Langevin equation, 0 f 8S ~ 8K(x,x';¢) f Ot ¢ ( x ' t ) = - ddx'K(x'x';¢) 5¢(x',t~ + ddx' 5V(x',t) + ddx'G(x'x';¢)q(x"t)' (1.1a) where t is the fictitious-time variable, and the kernel K(x, x'; 0) is a real functional depending only on ¢ at fictitious time t. The gaussian white noise q(x, t) is characterized by the statistical properties, (q(x,t))=O, (q(x,t)q(x',t'))=2da(x-x')3(t-t '). (1.1b) ~ Ref. [9 ] includes some arguments of stabilizingstochastic processes using a kernel. 0370-2693/92/$ 05.00 © 1992 ElsevierSciencePublishers B.V. All rights reserved. 129