All Questions
25
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
76
views
physical interpratation of partition function in Quantym field theory
Partition function in Statistical mechanics is given by
$$ Z = \sum_ne^{-\beta E_n} $$
For QFT, it is defined in terms of a path integral:
$$ Z = \int D\phi e^{-S[\phi]} $$
How can we see the relation ...
- 411
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
4
votes
0
answers
60
views
Thermodynamic free energy of interacting system
This question concerns an interacting system's thermodynamic free energy $\Omega$. Generally speaking, The action $S$ for an interacting system has the following form:
\begin{equation}
S(\phi,\psi) = ...
- 2,038
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
1
vote
1
answer
83
views
Clarification regarding the terminology of Microstates
I would like to understand how microstates are defined or used in physics. Are microstates suppose to only mean eigenvalues of a given observable (or a generator of symmetry)? The reason for my ...
- 1,200
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
0
votes
0
answers
311
views
Free Energy vs. Partition Function in QFT
The partition function of QFT is defined as
$$Z=\int\mathcal{D}\varphi e^{iS[\varphi]}.$$
Now, it is a general fact that this formal path integral can be computed perturbatively as (sketchy)
$$Z=\sum_{...
- 854
1
vote
0
answers
248
views
Quantum to classical mapping
I'm having troubles understanding precisely how the mapping from a quantum system to a classical one works.
Let's say that I have a quantum system in $d$ dimensions with Hamiltonian $H$ at temperature ...
- 741
4
votes
1
answer
424
views
In QFT, why are the vacuum partition function and the zero-temperature imaginary-time partition function the same?
When doing thermal field theory, one can start with the definition of the (thermal) partition function $Z = Tr[e^{-\beta H}]$, and inserting a number of completness-relations, we can arrive at (I am ...
- 577
4
votes
3
answers
523
views
The full path integral of a quantum field theory
Suppose if one is able to do a full path integral of a QFT with an action say $S[\phi]$ i.e.
$$Z = \int [\mathcal{D}\phi] e^{iS[\phi]}.$$ What can I use $Z$ for? Can I use the $Z$ like the partition ...
- 1,200
1
vote
0
answers
61
views
Derivation of fermionic partition function, how does commutation work?
When deriving the fermionic partition function with coherent states $|\psi\rangle$ we make the following step
$$
\mathcal Z=\int d(\bar\psi,\psi)\ e^{-\sum_i\bar\psi_i \psi_i}\sum_n\langle n|\psi\...
- 23
1
vote
0
answers
153
views
Path integrals on classical statistical mechanics
I'm learning a little bit about path integrals by myself lately and notice something quick curious. So far, I've learned that path integrals have many applications in physics, including quantum ...
- 585
-1
votes
1
answer
379
views
Partition function in quantum field theory
Why does the partition function include current term in free scalar field
$$Z[J] = \int \mathcal{D}\phi \, e^{i \left(S[\phi] + \int d^4x \,J(x) \phi(x) \right)}~$$
- 39
1
vote
0
answers
256
views
What is the entropy and/or equation of state of a partition function such as $Z=\int D\phi \exp (i S[\phi]/\hbar)$?
At this link https://en.wikipedia.org/wiki/Partition_function_(mathematics), it is claimed that the following partition function:
$$
Z=\int D\phi \exp (-\beta H[\phi]) \tag{1}
$$
is a consequence of ...
- 1,548