Questions tagged [grassmann-numbers]
341 questions
34
votes
5 answers
"Velvet way" to Grassmann numbers
In my opinion, the Grassmann number "apparatus" is one of the least intuitive things in modern physics.
I remember that it took a lot of effort when I was studying this. The problem was not in the algebraic manipulations themselves -- it was rather…
Kostya
- 20,288
30
votes
3 answers
Grassmann paradox weirdness
I'm running into an annoying problem I am unable to resolve, although a friend has given me some guidance as to how the resolution might come about. Hopefully someone on here knows the answer.
It is known that a superfunction (as a function of…
QuantumDot
- 6,531
26
votes
1 answer
Classical Fermion and Grassmann number
In the theory of relativistic wave equations, we derive the Dirac equation and Klein-Gordon equation by using representation theory of Poincare algebra.
For example, in this paper
http://arxiv.org/abs/0809.4942
the Dirac equation in momentum space…
Xiaoyi Jing
- 1,110
21
votes
3 answers
Why cannot fermions have non-zero vacuum expectation value?
In quantum field theory, scalar can take non-zero vacuum expectation value (vev). And this way they break symmetry of the Lagrangian. Now my question is what will happen if the fermions in the theory take non-zero vacuum expectation value? What…
Paul
- 361
15
votes
0 answers
Where does a fermionic coherent state live (which Hilbert space)?
There have been a couple of questions on fermionic coherent states, but I didn't find any that covered the following question:
If I define a coherent fermionic state in the 2-level-system spanned by $|0\rangle$ and $|1\rangle$, I will write it…
LFH
- 760
15
votes
1 answer
Why is there extra minus sign in Feynman's rules for every closed fermionic loop?
I know this is connected to the fact that fermions are represented by anticommuting operators, but I still cannot find the way to get this minus in Feynman rules.
Newman
- 2,626
15
votes
1 answer
The correspondence between Grassmann number and 4-spinor
In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number.
For two Grassmann number $\psi_1$, $\psi_2$, we can…
346699
- 6,211
14
votes
1 answer
Do composite-boson-states live in symmetrized or antisymmetrized Fock Space?
An even number of fermions make up a boson. Is this state described by a vector in antisymmetrized (fermionic) Fock space, where the resulting vector can somehow be connected to a bosonic state, or is it empirically motivated that an even number of…
Michiel
- 73
14
votes
1 answer
What is the value of a quantum field?
As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point in space. These then act on the vacuum or…
Siraj R Khan
- 2,028
13
votes
1 answer
Can change in position due to acceleration be expressed using dual quaternions?
Dual quaternions seem like an appealing way to model 6DOF motion since they linearize rotation.
I've reviewed what literature I can find on then, and found expressions for translation and change in position for constant velocity, but not for…
gibbss
- 183
13
votes
1 answer
Basic Grassmann/Berezin Integral Question
Is there a reason why $\int\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\! d\theta = 0$ which I can follow, but not for the former one.
F R
- 131
11
votes
1 answer
Integrals over Grassmann numbers
I want to prove an identity from Peskin&Schroeder, namely that
$$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m \theta_l^* \exp(\theta_j^* B_{jk} \theta_k)=\det(B) B^{-1}_{ml}\tag{9.70}$$
$B$ is a hermitean $N\times N$ matrix…
TheoPhysicae
- 397
11
votes
2 answers
Path integral for complex scalar field
I am taking a QFT course which focuses on the path integral formulation.
At a certain point, I was confused because we saw that, when integrating over complex Grassmann fields for fermions, we defined the complex conjugate as
$$(\theta\eta)^* =…
Marcosko
- 392
11
votes
1 answer
Assumptions of the Coleman-Mandula Theorem
In the original paper All Possible Symmetries of the S-Matrix, by S. Coleman and J. Mandula, they prove their famous 'no go' theorem regarding the possible extensions of Poincaré symmetry. The loophole that allows for supersymmetry is their…
JamalS
- 19,600
10
votes
2 answers
Do derivatives anticommute with Grassmann variables and complex numbers in a many-body path integral?
I'm trying to learn how to do a many-body path integral for both fermions and bosons, and I'm stuck. I'm following Altland and Simons - Condensed Matter Field Theory, chapter 4. On page 167, equation 4.27 is
\begin{equation}
Z = \int \prod_{n=1}^N…
Jordan
- 1,663