All Questions
20
questions
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
0
votes
0
answers
43
views
Calculation about fermions via quantum field theory
I want to ask a specific question occurred in my current learning about neutrinos.
What I want to calculate is an amplititude:
\begin{equation}
\langle\Omega|a_{\bf k m}a_{\bf pj}a_{\bf qi}^{\dagger}...
- 1
1
vote
0
answers
34
views
Normalisation for a two fermion state
I'm trying to follow this paper (Fermion and boson beam-splitter statistics. Rodney Loudon. (1998). Phys. Rev. A 58, 4904)
However, I don't quite understand where some of his results come from.
...
- 33
5
votes
2
answers
336
views
Grassmann numbers for fermions in QFT
I'm studying the Grassmann variables from Polchinski's string theory textbook appendix A. On page 342,
In order to follow the bosonic discussion as closely as possible, it is useful to define states ...
- 1,783
4
votes
2
answers
234
views
How to show $\frac{\delta}{\delta \psi(x)}$ being a representation of the operator $\Psi(x)^{\dagger}$ for fermionic Schroedinger Functionals?
I'm following the book of Brian Hatfield, Quantum Field Theory of particles and strings, page 217, eq. 10.89 and the following.
The author is looking for a representation of the operators $\Psi(x)$ ...
- 6,763
3
votes
1
answer
169
views
By using a Hilbert space (enhanced by Grassmann Numbers), can we write down a full set of eigenstates of the fermionic field operator?
By extending the Hilbert space, using grassmann numbers instead of complex numbers, we can write down eigenstates of the fermionic annihilation operator $a$ without getting into trouble with the ...
- 6,763
2
votes
0
answers
77
views
Why does normal-ordering ensure finiteness?
I will be using Jan von Delft's rigorous construction of bosonization/refermionization as an example, but I will try to explain my question in more general terms.
Consider an (countably) infinite-dim (...
- 2,123
4
votes
1
answer
282
views
How do we trace over subregions in a fermionic QFT?
Bosonic Case
In a bosonic QFT, the Hilbert space associated to a surface $\Sigma$ is the appropriate space of wavefunctionals on $\Sigma$. Hence, if $\Sigma=\Sigma_1 \sqcup \Sigma_2$, we find that the ...
- 1,455
1
vote
1
answer
182
views
Requirement of Jordan-Wigner string in creation operator on Fock state
Our lecture notes described the action of the particle creation operator on a fermionic Fock state:
$$c_l^\dagger |n_1 n_2...\rangle = (-1)^{\sum_{j=1}^{l-1}n_j}|n_1 n_2 ... n_l+1 ...\rangle.$$
I am ...
- 2,604
0
votes
1
answer
207
views
Two-site fermion system
I've to study a two-site fermion system with hamiltonian
$$H=\sum_{\sigma=\uparrow,\downarrow}[\epsilon_1 c^+_{1\sigma}c_{1\sigma}+\epsilon_2 c^+_{2\sigma}c_{2\sigma}+w(c^+_{1\sigma}c_{2\sigma}+c^+_{2\...
- 13
2
votes
1
answer
525
views
What is a fermionic field theory?
Let $\mathscr{H}$ be a Hilbert space and $\mathscr{H}^{n}$ be the associated $n$-fold tensor product of this Hilbert space. I'll skip the mathematical details in what follows, but my approach follows ...
- 585
3
votes
1
answer
722
views
Minus Sign in Fermionic Creation and Annihilation Operators
I have the same question as the person here:
Action of Fermionic Creation and Annihilation Operators
The question actually wasn't anwered, because using anticommutation relations between creation $c_\...
- 182
2
votes
1
answer
354
views
Is the expectation value of a Fermi field operator a Grassmann number?
It's often noted that Bosonic fields result from quantizing classical field theories defined on a regular numbers, whereas Fermionic fields arise when quantizing a classical field theory defined on ...
- 21
6
votes
0
answers
189
views
What's the momentum-space vacuum wave-functional of a fermion?
In the Schrödinger picture, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\phi(\mathbf x)|\phi\...
- 970
6
votes
1
answer
573
views
What's the ground state wave-functional of a fermion?
The vacuum state, free field wave-functional of a scalar field $\hat\phi(x)$ in the Schrödinger representation of quantum field theory is
$$\begin{array}{cl}
\Psi_0[\phi] &= C\prod_k e^{-\omega(k)...
- 970