All Questions
70
questions
0
votes
1
answer
73
views
$2\pi$-rotation of fermionic states vs. fermionic operators
Given a fermionic state $|\Psi\rangle$, a $2\pi$ rotation should transform it as
\begin{equation}
|\Psi\rangle \quad\to\quad -|\Psi\rangle \,,
\end{equation}
On the other hand, given a fermionic ...
- 426
3
votes
2
answers
377
views
Proving a Grassmann integral identity
How to prove the following identity
$$
\begin{align}
\int {\rm d} \eta_{1} {\rm d} \bar{\eta}_{1} \exp\left(a \left(\bar{\eta}_{1}-\bar{\eta}_{0}\right)\left(\eta_{1}-\eta_{0}\right) + b \left(\bar{\...
- 682
2
votes
1
answer
88
views
Why reasonable observables are made of an even number of fermion fields?
On Michele Maggiore book on QFT (page 91) is stated, out of nothing, that "observables are made of an even number of fermionic operator" and similar sentences is in Peskin book (page 56).
Is ...
- 613
5
votes
1
answer
439
views
Dirac Lagrangian in Classical Field Theory with Grassmann numbers
The concept of the Grassmann number makes me confused.
It is used to describe fermionic fields, especially path integral quantization.
Also, it is used to deal with the classical field theory of ...
- 79
1
vote
1
answer
80
views
Path Integral Measure Transformation as $(DetU)^{-1}$
The path integral measure transforms as $D\Psi\rightarrow (DetU)^{-1}D\Psi$ for fermions, with $DetU=J$ the Jacobian.
I am referring to Peskin and Schroeder's Introduction to Quantum Field Theory, ...
- 57
2
votes
1
answer
144
views
Schwartz's Quantum field theory (14.100)
I am reading the Schwartz's Quantum field theory, p.269~p.272 ( 14.6 Fermionic path integral ) and some question arises.
In section 14.6, Fermionic path integral, p.272, $(14.100)$, he states that
$$ ...
- 323
2
votes
1
answer
121
views
Why does fermion have the expansion with Grassmann-numbers?
I learn the chiral anomaly by Fujikawa method. The text book "Path Integrals and Quantum Anomalies, Kazuo Fujikawa", in the page 151, says that
…one can define a complete orthonormal set $\{...
- 129
3
votes
1
answer
323
views
Classical fermions, where are they?
Context:
Studying the path integral formulation of QFT I stumbled upon a fairly simple statement: when doing loop expansions of a partition function:
$$Z[\eta ; \bar{\eta}] = \int [d\psi][d\bar{\psi}]...
- 1,611
1
vote
1
answer
108
views
When does the spinor need to be in a Grassmann variable?
Follow the closed question When does the spinor need to be in a grassmann variable?
--
Does the spinor in the spinor representation of the space-time symmetry
Lorentz space-time symmetry, like $so(1,...
- 593
2
votes
1
answer
427
views
What are self-interacting fermions?
There're a bunch of models of fermions with quartic self-interactions. There's an introduction from this wikipedia page.
For example, one can construct the Soler model of self-interacting Dirac ...
- 2,923
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
2
votes
1
answer
167
views
Jacobian functional matrix for fermionic path integral
I am revisiting Srednicki's book Chapter 77 and struggling to understand how you define the change of variables in the fermionic field integral
Srednicki defines the Jacobian functional matrix for the ...
- 143
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
3
votes
2
answers
239
views
Connection between column matrix and Grassmann numbers in Dirac field
In canonical quantization the Dirac equation is a complex column matrix, while in path integral formulation it's Grassmann numbers.
Is there a formula to convert from complex matrix to Grassmann ...
- 925
3
votes
1
answer
178
views
What Object is the Dirac Lagrangian in the functional treatment of QFT, where $\Psi$ and $\bar{\Psi}$ are Grassmann-numbers?
As far as I understood, in the path integral formulation of QFT, a field configuration is modelled by a mapping
$$
x \rightarrow \Psi(x)
$$
Where $\Psi(x)$ are 4 components, each represented by 4 ...
- 6,763