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Stochastic Quantization, Reflection Positivity, and Quantum Fields

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Abstract

We investigate stochastic quantization as a mathematical tool for quantum field theory. We test the method for the free scalar field. We find that the usual method of stochastic quantization is incompatible with establishing a Hilbert-space interpretation for transition probabilities in quantum theory. In particular, we prove that for any finite stochastic time, the standard probability measure violates reflection positivity. As a consequence, if one desires to use stochastic quantization in constructive quantum field theory, one needs to find a more robust procedure than the standard one.

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Notes

  1. I am grateful to Alex Wozniakowski for assisting me to use Mathematica to test whether F(t) changes sign.

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Acknowledgments

This work was supported in part by a grant from the Templeton Religion Trust.

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  1. Harvard University, Cambridge, MA, 02138, USA

    Arthur Jaffe

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  1. Arthur Jaffe
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Jaffe, A. Stochastic Quantization, Reflection Positivity, and Quantum Fields. J Stat Phys 161, 1–15 (2015). https://doi.org/10.1007/s10955-015-1320-z

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