# Questions tagged [anticommutator]

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### What is the motivation for the Self-Dual Canonical Anticommutation Relation (CAR) algebra?

What exactly is the motivation for the use of the Self-Dual Canonical Anticommutation Relation (CAR) algebra in the context of infinite lattice systems? Why not remain with the CAR algebra, as both ...
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### Why can the commutator of a general expression be replaced by the anti-commutator in the $bc$ CFT theory?

Polchinski states in his equation 2.6.14 (in his book String Theory Vol. 1, Introduction to the Bosonic String) that for charges $Q_1$ and $Q_2$ the following equation holds, where $j_i$ is the ...
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### Can Hadamard's formula be used for fermionic operators?

Can I use this special case of Hadamard's formula $$e^\hat B \hat A e^{-\hat B}= A + [B,A]+\frac{1}{2!}[B, [B,A]] + \dots$$ for fermionic operators? Suppose I have fermionic operators that obey ...
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1 vote
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### Adding a surface term to Dirac action modifies the canonical anticommutation relations [duplicate]

I'm dealing with the following issue: when describing a fermionic field, one can use the typical Dirac Lagrangian $$\mathcal{L}=\bar\psi(i\gamma^\mu\partial_\mu-m)\psi,\tag{1}$$ or the more symmetric ...
• 503
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### Why do fermions anti-commute in Ising model?

In my course fermions are given like a product of spin and (dual to spin) disorder parameter in 2D Ising square lattice. Then, using the properties of disorder parameter I can prove that fermions ...
1 vote
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### Where do arbitrary phases of wavefunctions go under second-quantization?

As far as I understand, a second-quantized operator in QFT or condensed matter represents a many-body wavefunction (symmetrized for bosons or antisymmetrized for fermions). But every wavefunction is ...
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### Electron fields does not anticommute at space-like points

In the end of page 804 and beginning of page 805 of Streater's paper Outline of axiomatic relativistic quantum field theory which can be find here https://iopscience.iop.org/article/10.1088/0034-4885/...
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1 vote
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### Can the operator field Dirac equation be expressed as Heisenberg's equation?

The Dirac equation of the operator spinor field is: $$(i\gamma ^{\mu}\partial _{\mu} -m)\psi =0$$ where $\psi$ is interpreted to be a quantum field. I'm wondering, can this be derived from the ...
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### Anticommutation relations for fermionic operators in Fock space

In second quantization, creation and annihilation operators are defined on Fock space as follows: \begin{align} \begin{cases}a_j^\dagger|n_1,n_2,...,n_j,...\rangle=\xi^{s_j}\sqrt{n_j+1}|n_1,n_2,...,...
1 vote
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### Is there a Stone-von-Neumann theorem-like result for the canonical anti-commutation relations (CAR)?

The canonical commutation relation (CCR) $$[\phi(x), \pi(y)] = i\hbar\delta(x-y)$$ is kind of the key to basically any bosonic quantum theory. This is due to many different remarkable properties: By ...
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### Which of these commutation relations are correct? [closed]

I saw, in two different references, the following two commutation relations for the fermionic field operator: and which one of them is correct? 1 "Stefanucci, Gianluca, and Robert Van Leeuwen. ...
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### Anticommutation and Bogoliubove transformation

I am given the following transformation: \begin{bmatrix} ...
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### What is the physical meaning of the anticommutator of two observables? [duplicate]

It is quite clear to me that when two operators commute it implies that two different observables associated with the respective operators can be measured simultaneously with the exact accuracy. But ...
138 views

### Spinor index, dirac field equation

Sometimes I read anticommute $$\{\psi(x),\psi^\dagger(y)\}=\delta^{(3)}(x-y)$$ Sometimes, $$\{\psi_a(x),\psi_b^\dagger(y)\}=\delta^{(3)}(x-y)\delta_{ab}$$ Are they the same, second one just emphasis ...
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### Are there any 3 or more Hermitian solutions to the problem: $\alpha_i^2=1$, $\{\alpha_i, \alpha_j \}=2$

I’m trying to generate some matrices which are similar to Pauli’s but with the following anti-commutation relation $$\{\alpha_i, \alpha_j\}=\alpha_i \alpha_j + \alpha_j \alpha_i = 2 \tag{1}$$ And \...
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### Commuting but not anti-commuting operators

Two Hermitian operators $\hat{A}$ and $\hat{B}$ are such that they commute but don't anti-commute. In this case, even they commute their uncertainty product will not be zero, is it right?
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### Derivation of anti-commutation relations of massive supermultiplet generators [closed]

In almost all intro to supersymmetry notes the commutation relations are given between the generators and their conjugates however, I can not find any proofs of them anywhere and am struggling to ...
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### Defining particles by their commutation/anti-commutation relations

In my studies of many-body physics, I have encountered three types of particles, which can be defined based on their commutation/anti-commutation relations. Fermions, defined by raising/lowering ...
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