Temporal breakdown and Borel resummation in the complex Langevin method
We reexamine the Parisi-Klauder conjecture for complex e{sup i{theta}/2}{phi}{sup 4} measures with a Wick rotation angle 0{<=}{theta}/2{<=}{pi}/2 interpolating between Euclidean signature and Lorentzian signature. Our main result is that the asymptotics for short stochastic times t encapsulates information also about the equilibrium aspects. The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t{yields}0 asymptotic expansion which is shown to be Borel summable. The Borel transform correctly reproduces the time dependent moments of the complex measure for all t, including their t{yields}{infinity} equilibrium values. On the other hand the results of a direct numerical simulation of the Langevin moments are found to disagree from the 'correct' result for t larger than a finite t{sub c}. The breakdown time t{sub c} increases powerlike for decreasing strength of the noise's imaginary part but cannot be excluded to be finite for purely real noise. To ascertain the discrepancy we also compute the real equilibrium distribution for complex noise explicitly and verify that its moments differ from those obtained with the complex measure. - Highlights: Black-Right-Pointing-Pointer The Parisi-Klauder conjecture is reexamined for complex e{sup i{theta}/2}{phi}{sup 4} measures. Black-Right-Pointing-Pointer The time dependent moments are evaluated by temporal Borel resummation. Black-Right-Pointing-Pointer The results disagree with the Langevin simulations beyond a critical time t{sub c}. Black-Right-Pointing-Pointer t{sub c} increases with decreasing strength of the noise's imaginary part. Black-Right-Pointing-Pointer The technical reason for the breakdown is identified.
- OSTI ID:
- 22157069
- Journal Information:
- Annals of Physics (New York), Journal Name: Annals of Physics (New York) Vol. 329; ISSN 0003-4916; ISSN APNYA6
- Country of Publication:
- United States
- Language:
- English
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