# Questions tagged [grassmann-numbers]

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### Grassmann parameter in supersymmetry

Let's consider a free Wess-Zumino Lagrangian given by $$\mathcal{L} = \partial^{\mu}\overline{\phi}\partial_{\mu}\phi + i\psi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi\tag{1}$$ Whose ... 56 views

1 vote
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### Grassmann numbers and fermionic strings

Is it correct that by introducing Grassmann numbers as new directions of spacetime we can make strings behave like fermions (that is, 1/2-spin objects)? And if so, is it possible to show how that ...
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### 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 ...
1 vote
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### What is the problem with classical fermionic field?

Consider classical fermionic field. We have it's action, equations of motion and so we can get it's solutions, right? For example, we can consider gravitational solutions with fermions (in particular, ... 174 views

### Application of "real" Grassmann Gaussian integrals

In Appendix 2B of the CFT yellow book by Francesco et al, the authors introduced two types of Grassmann Gaussian integrals (the $\theta$'s below are generators of a Grassmann algebra): The "real&...
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### Born's Rule for states over supernumbers?

For Quantum-mechanics on a Hilbert-space over the complex numbers, the usual scalar product of two states $\langle \phi | \psi \rangle$ and gives the transition amplitude between the two states. The ...
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### Gaussian Grassmann integral with complex/bosonic source term

I'm interested in solving the following multi-dimensional integral $$\int d \theta d \bar{\theta} e^{-\bar{\theta}M \theta +\Lambda \theta + \bar{\theta} J }$$ where $\theta$ is a $N$-dimensional ...
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### How does the Pauli Exclusion Principle work in Twistor theory?

Twistor theory is described sometimes as a 'natural' way to represent spinor fields. In QED, we have the Grassmann valued spinor field $\Psi^\alpha(x)$, which naturally leads to the exclusion ...
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### Is it wrong to say that nature is in a superposition of fermionic coherent states?

We can't observe grassmann numbers in nature, and physical systems "in nature" are never in a fermionic coherent state (whose eigenvalues are grassmann numbers). However, if for example in ...
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### Are Grassmann numbers always "under the hood" if we deal with fermionic ladder / field operators?

For the set of all fermionic field operators $\Psi(x) | x \in \mathbb{R}^{3 +1}$, we won't find a $|\phi \rangle$ that is an eigenstate to the complete set of field operators, unless we make use of ...
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### 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 ...
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### What does it exactly mean by right and left functional derivatives?

In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions. I do not quite see what it means by that. For example, let us ...
1 vote
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### Anti-Symmetry of Dirac Operator

In his paper Fermion Path Integrals And Topological Phases, Witten states “Whenever one has a theory of fermions, the quadratic part of the fermion action is always antisymmetric by virtue of fermi ...
1 vote
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### Feynman rules in Grassmann variables [duplicate]

I'm given the following integral  Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2}...
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### Complex valued Grassmann variables $(\theta \eta)^*$, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$

Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom. We start by accepting the ...
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### 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 ...
1 vote
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 ...