Questions tagged [anticommutator]

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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{equation} \begin{bmatrix} ...
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Anticommutator of Fourier transform

Let \begin{equation} a_x = \int_{-\pi}^\pi \frac{\text{d}q}{2\pi}e^{iqx}a(q) \end{equation} \begin{equation} a_x^\dagger = \int_{-\pi}^\pi \frac{\text{d}q}{2\pi}e^{-iqx}a^\dagger(q) \end{equation} ...
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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 ...
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2 votes
1 answer
75 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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1 vote
0 answers
85 views

Do fermionic creation/annihilation anticommutation relations fix the creation and annihilation operators?

If you define operators $a, a^\dagger$ which satisfy (e.g.) the relations $\{a,a\}=\{a^\dagger,a^\dagger\}=0$ and $\{a,a^\dagger\}=1$. Will this uniquely define the operators such that $a |0\rangle \...
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2 answers
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Electron pair operators and anticommutation relations [closed]

Fermions creation and annihilation operators obey the anti-commutation relations given by: $$ \{ c_{\bf{k} \sigma} , c^{+}_{\bf{k'} \sigma'} \} = \delta_{\bf{k}\bf{k'}}\delta_{\sigma\sigma'} \\ \{ c_{...
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Anticommutation relation for the Weyl spinors in Minkowski space+time

For $D$-dimensional Minkowski space+time, I suppose that the Dirac spinor has the following anticommutation relation: $$ \{ \psi(x_1), \psi(x_2)\}=\{ \psi^\dagger(x_1), \psi^\dagger(x_2)\}=0 $$ $$ \{ \...
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Anti-commutation relation for gamma matrices; when/why did the definition change?

The anti-commutation relation defining gamma matrices is presently given by $$\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2g^{\mu\nu}$$ where appears the matric tensor (see for instance the ...
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From Fermions to CAR

I do not understand why Fermions statistic is equivalent to the CAR. In other words, suppose we have $\{a_i\}_i$ set of Fermionic operators acting on an Hilbert space, they satisfy the Canonical ...
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How should I imagine a spinor commutator, or consecutive occurences of $\bar{\Psi}$ and $\Psi$ in general?

I'm having a hard time making sense of an expression like $$\left[\Psi(x), \bar{\Psi}(y)\right].$$ Up until now I imagined a spinor operator to be something like a column vector of operators, ...
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2 votes
1 answer
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Proof involving exponential of anticommuting operators

Problem: On page 23 of the book "Quarks, gluons and lattices" by Creutz, he defines a state $$\langle\psi|=\langle 0|e^{bFc}e^{\lambda b^\dagger G c^\dagger}$$ where $\lambda$ is a number, $...
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1 answer
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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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Super Virasoro Algerbra in string theory [closed]

I'd like to calculate the algebra ${G_r,G_s}$ for the Ramond sector, or Neuveu-Schwarz Sector for super Strings. The problem is that I don't know how to evaluate the anti-commutator relation between ...
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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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4 votes
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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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2 votes
1 answer
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Right derivative of Grassmann number and associated anti-commutation relation

I am reading chapter 3 of Anomalies in quantum field theory by Reinhold Bertlmann and I found one statement that I don't know how to prove. First of all he defined the right derivative on the ...
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On the normal ordering of Fermi fields

From my understanding, the normal ordering of Klein-Gordon fields in QFT is valid because of the ambiguity that comes with quantizing a classical theory, in the sense that the conmutator of fields is ...
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Pauli Exclusion Princple for a fermion and antifermion

I understand that the Pauli Exclusion Principle applies only for identical particles, so that a fermion and an anti-fermion should be allowed to be in the same state. However, when I look at the ...
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1 answer
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Anti-commutator of Dirac matrices

Consider $$ \beta = \begin{pmatrix} \mathbf{1} & 0 \\ 0 & -\mathbf{1} \end{pmatrix},\quad \alpha_i = \begin{pmatrix} 0 & \mathbf{\sigma}_i \\ \mathbf{\sigma}_i&0 \end{pmatrix}.$$ The ...
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(Anti)commutation of creation and annhilation operators for different fermion fields

The Fourier expansion of the fermion field operator is such that $$ \hat\psi=\int\!d^3p\,\left[ f_b(p)\hat b(p) +f_d(p)\hat d^\dagger\!(p) \right] ~~, $$ for some sufficiently complicated $f_b$ and $...
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1 vote
1 answer
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Measurements, QFT and Wightman's axiom 3

I think I might have misconceptions about the conceptual core of QFT. Let me explain where I am puzzled. In QM, the measurement process is accounted by the postulate of collapse of the wave function: ...
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Show $[\gamma^\mu,\eta^{\nu\lambda}I]=0$ for Dirac matrices

I am trying to show the following commutation relation for the Dirac matrices $\gamma^\mu$ and the metric $\eta^{\nu\lambda}$: $$2[\gamma^\mu,\eta^{\nu\lambda}I]=0$$ where $I$ is the 4x4 identity ...
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2 votes
1 answer
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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 ...
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3 votes
1 answer
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Why is the anti-commutation relation $\lbrace \psi_a(x), \bar{\psi}_b(y) \rbrace = 0$ enough to ensure causality?

In quantum field theory, it is crutial that two experiments can not effect each other at space-like seperation. Thus $[\mathcal{O}_1(x), \mathcal{O}_2(y)] = 0 $ if $(x-y)^2 < 0$. For the Klein-...
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3 votes
1 answer
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Why is the anticommutator of the uncertainty principle omitted if it serves to increase the accuracy of our "knowledge" of a quantum state?

The generalized uncertainty principle can be derived and shown to be this which is fine and rigorous. $\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle ...
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1 answer
107 views

How to express the anti-commutator in the form of a density operator?

$ \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} \newcommand{\braket}[2]{\langle{#1}|{#2}\rangle} \newcommand{\acomm}[2]{\left\{#1,#2\right\}} $Let $\{ \ket{1} \ket{1} \ket{2} .....
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2 votes
2 answers
139 views

How do fermionic operators transform?

In quantum mechanics, if we have an operator $\Omega$, then under the transformation $T$, with infinitesimal generator $G$ (i.e. $T(\epsilon)=1-i\epsilon G + \ldots$), then operator transforms as $$\...
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Creation and annihilation operators for fermions from anticommutator

In a question, I was given that $a^{\dagger}a + a a^{\dagger} =1$ and asked to show what $a|n\rangle$ and $a^{\dagger}|n\rangle$ would be, given that $H|n\rangle=(a^{\dagger}a + 1/2)$. I am getting ...
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(Anti)commutators at different times

Why does the commutator of two operators evaluated at different times vanish? Take for instance a fermonic field $\psi_\sigma (\vec{x},t)$, which satisfies the well known anti-commutation relations ...
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Anticommutator of gauge covariant derivatives

I must convert some dimension-6 operators I've obtained to the SILH base (ref: this, "Review of the SILH basis", CERN presentation by R. Contino). In this conversion I've got operators such ...
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1 answer
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(Anti)commutation relations for higher-dimensional anti-symmetrized Gamma matrices

Suppose $a,b,c...=0,....,D-1$ are Lorentz indices of $SO(1,D-1)$ tangent space and consider $D$-dimensional Clifford algebra defined by the usual anticommutation relation $$\{\Gamma^a,\Gamma^b\}=2\eta^...
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3 votes
1 answer
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How to solve differential equation involving commutator and anti-commutator?

In one of my exercise, I got following differential equation for density matrix $\rho$, $$ \frac{d\rho}{dt}=-i[H_1,\rho]+\{H_2,\rho\} $$ where $H_1$ and $H_2$ are the Hermitian Hamiltonian, and $[.,.]$...
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6 votes
4 answers
445 views

How do you experimentally realise the operator of the form $\hat{A}\hat{B}+\hat{B}\hat{A}$?

I was reading a paper in which the authors use the operator of the form $\hat{A}\hat{B}+\hat{B}\hat{A}$ and it is implied to be experimentally realisable. (i.e either creating an apparatus to ...
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1 vote
0 answers
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Is there a bosonic representation of Clifford algebra in (1,3) spacetime?

By a suitable combination of Dirac's $\gamma_\mu$ matrices one can define creation and destruction operators satisfying fermionic anticommutators. Is there a similar result for bosons in the context ...
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1 vote
1 answer
50 views

Finding chiral-symmetric degenerate states numerically

I am dealing with a Chiral-symmetric Hamiltonian such that $$ 𝑆𝐻𝑆^{−1}=−𝐻. $$ Two of its eigenstates have zero eigenvalue and fulfill $𝑆∣𝜓_{\alpha}⟩=𝑒^{𝑖𝜙_{\alpha}}∣𝜓_{\alpha}⟩$, while the ...
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0 votes
1 answer
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Trick with "functional" derivative to evaluate commutators between diagonal hamiltonian and creation fermionic operator

I found a theorem that states that if $A$ and $B$ are 2 endomorphism that satisfies $[A,[A,B]]=[B,[A,B]]=0$ then $[A,F(B)]=[A,B]F'(B)=[A,B]\frac{\partial F(B)}{\partial B}$. Now i'm trying to apply ...
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5 votes
3 answers
2k views

Is there an identity for anti-commutator $\{ A B, \, C D \}$ in terms of commutators $[\, , \,]$ only?

I'm looking for an identity that could express the anti-commutator $$\tag{1} \{ A B , \, C D \} \equiv A B C D + C D A B $$ expressed as a combination of commutators only: $[A,\, C]$, $[A, \, D]$, etc....
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1 answer
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Anti-commutator for annihilation and creation operators: ordering of indices

I'm trying to prove that $\{\tilde a_i,\tilde a_j^{\dagger} \}=\delta_{ij}$, by defining $\tilde a_i=\sum_j \bar U_{ji}a_j$. U is an unitary matrix and $a_i$ refers to an element of the operator $a$. ...
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3 votes
1 answer
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$\mathcal{N} \ge 2$ Supersymmetry massive supermultiplets

In Bertolinis SUSY notes [https://people.sissa.it/~bertmat/susycourse.pdf] we have defined: $$ \{Q^I_\alpha,\bar{Q}_\dot{\beta}^J\}=2m\delta_{\alpha\dot{\beta}}\delta^{IJ}\tag{3.24} $$ $$ \{Q^I_\alpha,...
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1 vote
1 answer
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Chiral Symmetry and Charge Algebra

I'm following Section 5.1 in Cheng and Li's Particle Physics book and I am having trouble reproducing some of the commutation relations. The axial current is given by $$ (J_A^a)^\mu = \bar{\psi}_\...
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0 votes
1 answer
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Importance of position of Bosonic and Fermionic operators in quantum mechanics

In quantum mechanics if (Fermionic or Bosonic) operators do not commute with each other, one cannot swap position of two operators easily. For example, let $(c^\dagger, c)$ are Fermionic operators, ...
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3 votes
1 answer
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Connection between Poisson bracket and Anti-commutator?

Canonical quantization promotes Poisson brackets in classical mechanics to commutators in quantum mechanics. Is there any classical counterpart similar to the Poisson bracket for the anticommutator?
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3 votes
2 answers
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Propagator for Dirac spinor field

I am currently trying to learn Quantum Field Theory through David Tong's notes which only talk about canonical quantisation for the scalar field and Dirac spinor field. In Chapter 2, the propagator ...
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1 answer
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Fierz identities and anticommutation relations

Let us consider the following term $$\bar\psi(p_1)M\psi(p_2) \bar\psi(p_3)N\psi(p_4)$$ According to Ortin's book (equation (D.65)), from the Fierz identity we would have something like $$\bar\psi(...
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1 answer
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Dirac spinor and field quantization

Can we simplify $$ Σ_s Σ_r [b_p^s u^s(p)\mathrm e^{ipx} (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} + (b_q^r)^†(u^r)^†(q)\mathrm e^{-iqy} b_p^s u^s(p)\mathrm e^{ipx}]\tag{1}$$ as $$Σ_sΣ_r[ \{b_p^s, (b_q^r)^†...
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Anti-commutator in Quantization of Dirac field

Can anyone explain while calculating $\left \{ \Psi, \Psi^\dagger \right \} $, set of equation 5.4 in david tong notes lead us to $$Σ_s Σ_r [b_p^s u^s(p)e^{ipx} b_q^r†u^r†(q)e^{-iqy}+ b_q^r †u^r†(q)e^{...
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  • 349
0 votes
1 answer
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How to prove the following identity of fermion creation and annihilation operators [closed]

Define $$M_{\theta} \equiv \exp\left[\theta \sum_s \left(d^{\dagger}(\vec{p},s)b(\vec{p},s) -b^{\dagger}(\vec{p},s)d(\vec{p},s)\right)\right],$$ where $\theta$ is a continuous real parameter. Show via ...
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