All Questions
40
questions
3
votes
3
answers
608
views
Path integral at large time
From the path integral of a QFT:
$$Z=\int D\phi e^{-S[\phi]}$$
What is a nice argument to say that when we study the theory at large time $T$, this behaves as:
$$ Z \to e^{-TE_0} $$
where $E_0$ is the ...
- 411
1
vote
0
answers
146
views
How is Wick rotation an analytic continuation?
Wick rotation is formally described by the transformation
$$t \mapsto it.$$
In many place it is stated more rigorously as an analytic continuation into imaginary time. I understand why we do it but ...
- 3,350
2
votes
1
answer
176
views
Examples of Path integral $\neq$ Partition function?
Are there any systems we know of whose partition function is not simply Wick rotation of the path integral? Does anyone know of any examples?
- 1,200
3
votes
1
answer
106
views
How can we use saddle point approximation for a bounce solution which is not even a strict local minimum of the Euclidean action?
In calculating the false vacuum decay, the main contribution to the imaginary energy part of the Euclidean path integral comes from the bounce solution. And we somehow apply saddle point approximation ...
- 619
1
vote
0
answers
61
views
Is the Euclidean generating functional $Z_{E}[J]$ identified with original Minkowskian generating functional $Z[J]$?
In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to ...
- 81
3
votes
1
answer
224
views
Path Integral for Unruh Effect
In derivation of Unruh effect, according to arxiv 2108.09188, we have
$$
\langle\phi_L|\exp(-\pi H)|\phi_R\rangle=\int_{\phi=\phi_R}^{\phi=\phi_L} D\phi e^{-S_E}\propto \int_{lower\space half\space ...
- 133
4
votes
0
answers
123
views
Correlators on the Euclidean section of a black hole
In the standard construction of the Euclidean section of a Schwarzschild black hole, we start with the exterior metric in Schwarzschild coordinates:
$$\tag{1} ds^2 = -(1-r_s/r)dt^2 + (1-r_s/r)^{-1}dr^...
- 1,455
3
votes
0
answers
73
views
Can the QFT path integral be re-expressed using a real, positive-definite function of the action? [duplicate]
This question is based on my rather shaky grasp of QFT, so if I'm missing a key concept then just let me know!
If you're deriving the Schrodinger equation from the path integral as Feynman did, then ...
- 2,475
1
vote
1
answer
108
views
Why is the Vacuum state got by a limit to imaginary time?
Given a (non-relativistic) propagator $K_t(A,B)$ giving the 'conditional amplitude' to go from state $B$ to state $A$ in time $t$, it is known that one can find the vacuum wavfunction by (independent ...
user84158
6
votes
0
answers
319
views
Schwinger-Keldysh contour and $i\epsilon$ prescription
In Tom Hartman's notes on path integrals, he describes the Schwinger-Keldysh (or "in-in") formalism for calculating vacuum correlators in QFT.
He explains that Lorentzian time-ordered vacuum ...
- 1,455
6
votes
1
answer
199
views
Does a $d$-dimensional stat-mech theory necessarily have a $(d-1)$-dimensional quantum theory equivalence?
A $d$-dimensional stat-mech theory on a lattice usually can be represented by a $d$-dimensional tensor network. Taking a row/slice of tensors ($M$ tensors or sites) as the transfer matrix (MPO in 2$d$ ...
- 356
5
votes
1
answer
281
views
Do the Ward identities contain contact terms in Euclidean QFT?
In derivations of the Ward identities, I have never seen the signature of spacetime explicitly specified, so I'd always assumed they hold regardless of signature. However, the argument below seems to ...
- 1,455
1
vote
0
answers
248
views
Quantum to classical mapping
I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works.
Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
- 741
5
votes
1
answer
504
views
How does the $+i\varepsilon$ prescription in the propagator comes from analytic continuation of the Euclidean 2-point function?
Let $S_0[\phi]$ be the action for a real Klein-Gordon field $$S_0[\phi]=\dfrac{1}{2}\int d^Dx \phi(x)(\Box-m^2)\phi(x)\tag{1}.$$
If we try to construct the generating functional $Z_0[j]$ we find that ...
- 36.4k
4
votes
1
answer
424
views
In QFT, why are the vacuum partition function and the zero-temperature imaginary-time partition function the same?
When doing thermal field theory, one can start with the definition of the (thermal) partition function $Z = Tr[e^{-\beta H}]$, and inserting a number of completness-relations, we can arrive at (I am ...
- 577