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2 votes

Time evolution of operators in the path integral formalism

If we limit ourselves for a moment to one-particle path integrals, then a path integral is essentially a matrix element of the evolution operator (I might be skipping some nuaces here): $$ G(x,t|x_0,...
Roger V.'s user avatar
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0 votes

What is the physical meaning of the normalization of the propagator in quantum mechanics?

It seems like the following may be a possible answer. It works for the free particle (and I think the harmonic oscillator as well), although I'm not sure how to prove it's correct in general. Let $$ \...
zeroknowledgeprover's user avatar
3 votes
Accepted

Why isn't the free particle particle a function of the absolute value of the difference of the time?

Time evolution in QM is unitary $U(t)^{\dagger}=U(-t)$, so the free propagator is only expected to be invariant under combined time-reversal $t\to -t$ and complex conjugation. See also this related ...
Qmechanic's user avatar
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2 votes

Derivation of Schrödinger equation in Feynman-Hibbs

Briefly speaking, it follows from dimensional analysis that higher-order terms ${\cal O}(\eta^{n\geq 3})$ will [after the Gaussian $\eta$-integration (4.5)] only produce higher-orders terms ${\cal O}(\...
Qmechanic's user avatar
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2 votes
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How does inserting an operator in the path integral change the equation of motion?

The missing conceptual point to appreciate is that when you are calculating a correlation function it can be useful to interpret the same expression in different ways. We wish to calculate the ...
SethK's user avatar
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0 votes

How does inserting an operator in the path integral change the equation of motion?

Yes, the Wilson line (2.42) is now part of the action in the path integral. Yes, the symmetry defect operator (SDO) (2.41) is now part of the action in the path integral. References: T.D. Brennan &...
Qmechanic's user avatar
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2 votes

Path integral at large time

I'm assuming you mean immaginary time $T$. In this case the path integral is equal to the partition function (with $T=\beta$), as others have noted. Then, let the spectral decomposition of $H$ be $$ ...
lcv's user avatar
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5 votes

Path integral at large time

Path integral and operator formalism correspondence states that for Euclidean time $t$, $$Z=\int D\phi e^{-S[\phi]}\leftrightarrow Z~=~{\rm Tr}_{\cal H}(e^{-t\hat{H}})$$ Fix an orthonormal basis $\{|n\...
Kutasov's user avatar
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6 votes

Path integral at large time

Briefly speaking, the standard argument is A correspondence between the path integral formalism and the operator formalism in Minkowskian signature with unitary time evolution; Wick rotate both ...
Qmechanic's user avatar
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1 vote
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2PI Effective Action from Double Legendre Transformation

It is difficult to find an explicit reference that really shows this property, but one can convince oneself from eq. (2.17) in PhysRevD.10.2428. Note here that the constant is the log of the free ...
M_kaj's user avatar
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1 vote
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A Limited Sense of Path Integral Respecting Classical EOM

Weinberg's claim (3.) essentially boils down to the statement that the path integral of a functional derivative at some instant $t_A$ vanishes as long as $t_A$ is away from the initial/final time and ...
Qmechanic's user avatar
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