All Questions
27
questions
1
vote
0
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90
views
In the path integral formulation of QFT, how do we get quantized particles out of a field?
Every QFT textbook starts by basically postulating that we have discrete states connected by creation and annihilation operators. In Quantum Mechanics, we started from a differential equation and ...
- 1,262
3
votes
1
answer
153
views
Equivalence of Schrödinger operator formalism and Path Integral Formulations for Scalar Field Theory
I'm exploring the deep connections between different formulations of quantum field theories and have a specific question about the equivalence between the Schrödinger representation and the path ...
- 1,548
2
votes
1
answer
284
views
How to derive the Fermion generating function formally from operator formalism?
The generating functionals for fermions is:
$$Z[\bar{\eta},\eta]=\int\mathcal{D}[\bar{\psi}(x)]\mathcal{D}[\psi(x)]e^{i\int d^4x
[\bar{\psi}(i\not \partial -m+i\varepsilon)\psi+\bar{\eta}\psi+\bar{\...
- 619
8
votes
2
answers
575
views
Why is the limiting operator in the CFT state-operator correspondence well-defined, and why is conformal symmetry necessary?
Consider a Euclidean CFT in radial quantisation, and let $S$ be the unit sphere centred on the origin. The state-operator correspondence says that any state $\Psi_S$ living on $S$ can be prepared by a ...
- 1,455
8
votes
4
answers
2k
views
How to connect the usage of the path integral in QFT to the usage in Quantum Mechanics?
In Quantum Mechanics, path integrals are used to calculate the matrix element:
$$
\langle x_1, t_1|x_2, t_2\rangle_J=\int
e^{i(S[x(t)]+\int\!J x(t))/\hbar} d[x(t) ].\tag{1}$$
If we naively try to ...
- 6,355
1
vote
1
answer
102
views
Handling the $\nabla \phi$ term in the Hamiltonian in a path integral
Let the scalar Hamiltonian be of the form $H = \int d^3x \left [\hat{\pi}^2 + (\nabla \hat{\phi})^2 + m^2\hat{\phi}^2 \right ]$.
We wish to evaluate the quantity $\langle \phi_f | \exp(-iHt) | \phi_i \...
- 1,023
1
vote
1
answer
83
views
Clarification regarding the terminology of Microstates
I would like to understand how microstates are defined or used in physics. Are microstates suppose to only mean eigenvalues of a given observable (or a generator of symmetry)? The reason for my ...
- 1,200
2
votes
2
answers
367
views
Partition function for bosons with path integral
In this book the partition function for bosons is defined in eq. 2.17 as:
$$Z=\mathrm{Tr}[e^{-\beta (H-\mu_i N_i)}]=\sum_a\int d\phi_a\langle\phi_a|e^{-\beta(H-\mu_i N_i)}|\phi_a\rangle$$
The ...
user255856
7
votes
3
answers
1k
views
Euler-Lagrangian equation of motion of quantum fields in QFT
A canonical way of doing quantum field theory is by starting with some Lagrangian, for example, that of free scalar field
$$L=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi-\frac{1}{2}m\phi^2$$
Then ...
- 442
6
votes
2
answers
1k
views
Expectation values in path integral formalism
In quantum field theory, it is often assumed that the expectation value $\langle A\rangle$ of an operator $A$ can be written in the path integral formalism in the following way:
$$
\langle A\rangle = \...
- 588
0
votes
1
answer
98
views
Functional representation of operators in second quantization
In path integral formalism, the operator $\hat{a}$ and $\hat{a}^\dagger$ represented by numbers $\alpha$ and $\bar{\alpha}$ according to $\hat{a}$|$\alpha$⟩=$\alpha$|$\alpha$⟩ and <$\alpha$|$\hat{a}...
- 57
4
votes
1
answer
486
views
Preparing States using path integral in QFT
I had some confusion about the idea of cutting the path integral to define states in quantum field theory. There are two versions which I have seen:
We do the path integral with an unspecified '...
- 1,182
2
votes
2
answers
484
views
Path integral identity
I am reading the Background Field Methods in the EPFL Lectures on GR as an EFT. The authors use this identity on Page 23, Equation (174):
$$
\mathcal{N}^{-1}\int\mathcal{D}\phi\,\mathcal{D}\phi^*\exp\{...
- 8,492
3
votes
1
answer
449
views
Vacuum energy of a free scalar field from path integral
My question has been asked two other times:
Spinor vacuum energy (misleading title) and
Vacuum Energy Calculation using Path Integral. I am not completely satisfied with the answers and it looks like ...
- 713
7
votes
1
answer
206
views
Choice of folliation in path integral
Assume we have a scalar field theory for a field $\phi$. Can we think of the Hilbert space as being spanned by states of the form $|\varphi\rangle$ for configurations $\varphi\in C^\infty(\mathbb{R}^3)...
- 3,915