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Questions tagged [anticommutator]

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42 views

Anticommutation relations for fermionic fields imply that Hamiltonian / Lagrangian can at most be linear?

Fermionic field operators do obey anticommutation relations, so for a chosen Field operator (and the field momentum), we have: $$ \{\Psi_a, \Psi_b\} = \{\pi_a, \pi_b\}= 0 $$ with the $\Psi_a$ being ...
3
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1answer
41 views

Generalized commutator/anticommutator via phase factor

We know that the commutator between two operators $A$ and $B$ reads $[A,B]_{-}=AB - BA$, while the anticommutator reads $[A,B]_+=AB + BA$. I am wondering if someone has ever used a generalized ...
1
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1answer
33 views

Matrix representation of the CAR for the fermionic degrees of freedom

The canonical anticommutation relations (CAR) for a fermionic degree of freedom can be written as follows: $$ a^2 = \left( a^{\dagger} \right) ^2 = 0, $$ $$ a a^{\dagger} + a^{\dagger} a = 1. $$ ...
1
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2answers
90 views

Anticommutator of spin-1 matrices

We know that in the spin-1/2 representation the anticommutation relation of the Pauli matrices is $\{\sigma_{a},\sigma_{b}\}=2\delta_{ab}I$. Does a similar relation hold for the spin-1 representation?
1
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1answer
98 views

Why are Grassmann variables the classical limit of fermions?

In many texts the anti-commutation relations for fermions are given as $$\{ \bar{\psi}^\alpha (\vec{x}), \psi^\beta(\vec{y}) \} = \delta^{\alpha\beta} \delta(\vec{x} - \vec{y})$$ $$\{ \psi^\alpha (\...
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0answers
34 views

Commutation relations in QFT [duplicate]

So I have just started learning QFT. So you take a classical field and turn the degrees of freedom into operators. All fine, just like normal quantum. However I am confused about the commutation ...
2
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0answers
75 views

How are anticommuting fields ($\eta \chi = -\chi \eta$) “forced upon us” by representation theory of $SO(d-1,1)$?

I would like to know if anticommuting fields (which physicists use as fermions) emerge naturally from the spin representation theory of $SO(d-1,1)$. Is the fact that spinor fields anticommute a ...
0
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1answer
25 views

Anticommutativity of an anticommutator of supercharges

In this paper, equation 38 gives the ${\cal N}=2$ Super-Poincare (extended with the central extension $\mathcal{Z}$). The anticommutation relation of the two different supercharges is given as: $$\{Q^...
0
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1answer
41 views

Anti-Commutator of derivatives of Grassmann variables

How do I evaluate the anti-commutator of $\frac{\partial}{\partial\chi}$ and $\frac{\partial}{\partial\eta}$ when both $\chi$ and $\eta$ are Grassmann variables?
1
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3answers
129 views

Why doesn't the anticommutator $\{x,p_x\}$ have an unique value?

The commutator of position and momentum, $[x,p_x]$, has a unique value given by $i\hbar$. Why doesn't the anticommutator $\{x,p_x\}$ also have a definite value?
0
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1answer
26 views

Symmetry in Fock-space 2-body interaction

The simplest two body interaction term for fermions is $$H = \sum_{ijkl} U_{ijkl} a_i^\dagger a_j^\dagger a_k a_l$$ and I'm trying to determine the symmetries on $U$. Unfortunately I keep getting ...
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3answers
107 views

Heisenberg Uncertainty Principle derivation question [closed]

So I'm rading Shankaar's book and got stuck in this place. $$ (\Delta \Omega)^{2}(\Delta \Lambda)^{2} \geq \frac{1}{4}\left\langle\psi\left|[\hat{\Omega}, \widehat{\Lambda}]_{+}\right| \psi\right\...
0
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0answers
70 views

What is the QFT state with two distinguishable fermions present?

I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$. The state describing a single electron is given ...
1
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1answer
51 views

Explicit quantization of free fermionic field

The canonical quantization of a scalar field $\phi(x)$ can explicitly be realized in the space of functionals in fields $\phi(\vec x)$ (here $\vec x$ is spacial variable) by operators \begin{eqnarray} ...
3
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1answer
113 views

Canonical Quantisation vs the Dirac Field, why does it even work?

Using the "Dirac Prescription", we can preserve the format of a differential equation in its QM form. If we define the canonical variables s.t. they have the same commutation relations times $i$ as ...
0
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1answer
60 views

Proving identity $\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu}$

In the lecture notes accompanying a course I'm following, it is stated that $$\DeclareMathOperator{\Tr}{Tr} \Tr\left[\gamma^{\mu}\gamma^{\nu}\right] = 4 \eta^{\mu\nu} $$ Yet when I try to prove this,...
1
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1answer
47 views

Commutation relations of symmetry generators in SUSY

It is well known that the generators $$ Q_\alpha = \frac{\partial}{\partial \theta^\alpha} - i \sigma^\mu_{\alpha \dot \beta} \bar{\theta}^\dot{\beta} \partial_\mu $$ and $$ \bar{Q}_\dot{\alpha} = -\...
0
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1answer
180 views

Anti-commutation relations in annihilation operators

It is claimed that $$\{c_\alpha,c_\beta \} = c_\alpha c_\beta + c_\beta c_\alpha = 0$$ where $c_\alpha$ and $c_\beta$ are the fermionic annihilation operators in second quantization. Why is that ...
1
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3answers
564 views

Is there an anticommutator relation for orbital angular momentum?

So I know that there are commutator relations for $L$ such as $[L_x,L_y] = i\hbar L_z$, but is there a relation for the anticommutator? For example, $L_xL_y + L_yL_x$?
1
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1answer
115 views

Anticommutation relation different specie/type of fermion

Suppose we have two distinct fermions, say $X$ is Dirac, $Y$ is Majorana, part of different irreps of some Gauge group (e.g. SM). Alternatively, consider a lepton $l=l_L+l_R$ and a Majorana neutrino $...
1
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2answers
146 views

How to prove the translation generator commutes with the spinors in SUSY algebra?

I was reading Modern Supersymmetry by John Terning, the book starts with SUSY algebra and says $$ \left[ P_{\mu} , Q_{\alpha} \right] = \left[ P_{\mu} , Q_{\alpha}^{\dagger} \right] = 0 $$ I am ...
2
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2answers
152 views

How to change a commutator of SUSY super-charges into an anti-commutator?

I would like to understand an apparently rather simple calculation which checks the closure of the Supersymmetry algebra via the commutator of 2 supersymmetric variations of the type: $$\delta \phi = ...
1
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1answer
68 views

Problems with anti-commutator between fermionic ladder operators

I am trying to build the fermionic coherent state formalism in conformance with the grassmann conventions used in the book "Mirror Symmetry", relation (9.20), where the fermionic integration is ...
2
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1answer
2k views

Angular and linear momentum operators' commutation

Do linear and angular momentum operators commute? If I use the canonical commutation relations I get that they commute. Say, for $x$-component, $[p_x, L_x] = p_x y p_z - y p_z p_x - p_x z p_y + z p_y ...
0
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0answers
54 views

Anticommutation relation for the exponential field of the bosonic field

In 1+1 dimensions, the massless KG equation has the general solution $$\phi(x,t)=\int_{-\infty}^{\infty}dp/(4\pi E_p)[a_pe^{i(px-E_pt)}+a^{\dagger}_pe^{-i(px-E_pt)}]$$ where $E_p^2=p^2$. The operator ...
1
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3answers
163 views

What does self-closing bra-ket mean in Robetson-Schrodinger Uncertainty Relation?

I was reading: https://en.wikipedia.org/wiki/Heisenberg%27s_uncertainty_principle#Robertson–Schrödinger_uncertainty_relations Where an inequality is presented: $$ \sigma_A \sigma_B = | \frac{1}{2} \...
3
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4answers
2k views

Proof of the Anti-Commutation Relation for Gamma Matrices from Dirac Equation

My textbook on QFT says that the Dirac equation can be used to show the following relation: $$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}$$ I have searched around and unable to find how to prove this ...
0
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1answer
61 views

Anticommutator expression

I need to show that this expression is contradictory. The is no more information is given for $\hat{b}$. $$\hat{b}^{\dagger}\hat{b}+\hat{b}\hat{b}^{\dagger}=-I$$
2
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1answer
353 views

Why must fermion fields anticommute and bosons commute?

Fermion fields must satisfy anticommutation relation. But why? I know that unless they anti-commute the Pauli exclusion principle cannot be satisfied. But is there some other deeper/fundamental ...
1
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1answer
230 views

What's the reasoning behind propagators definitions (specifically fermionic propagators)

I'm studying QFT by David Tong's lecture notes. When he discusses causility with real scalar fields, he defines the propagator as $$D(x-y)=\left\langle0\right|\phi(x)\phi(y)\left|0\right\rangle=\int\...
0
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1answer
66 views

Do half integer spin fields commute or anti-commute with spin integer fields?

What are the fundamental commutation/anti-commutation relations between half integer and integer spin fields? For instance, in QED do we have \begin{equation} [\psi(x),A^{\mu}(y)]=0 \end{equation} or \...
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0answers
112 views

Anti-commutator version of Zassenhaus formula

The Zassenhaus formula goes like $$ e^{t(X+Y)}= e^{tX}~ e^{tY} ~e^{-\frac{t^2}{2} [X,Y]} ~ e^{\frac{t^3}{6}(2[Y,[X,Y]]+ [X,[X,Y]] )} ~ e^{\frac{-t^4}{24}([[[X,Y],X],X] + 3[[[X,Y],X],Y] + 3[[[X,Y],Y],...
0
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1answer
362 views

Quantizing the Dirac field using commutation relations leads to an unbounded Hamiltonian?

If we were to quantize the Dirac field using commutation relations instead of anticommutation relations we would end up with the Hamiltonian $$ H = \int\frac{d^3p}{(2\pi)^3}E_p \sum_{s=1}^2 ...
4
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2answers
427 views

Relation between spinors and anticommutation relation of fermions

I read that the state of a pair of particles is the tensor product of the single states of both, and you will get a wavefunction with the parameters of both, if you swap the parameters you will get a ...
5
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2answers
526 views

Why must the Bogoliubov transform preserve anticommutation relations?

$\mathbf{Background}$: In my research I am studying the Ising model, expressed in terms of Jordan-Wigner fermions: $$ H = \sum_{j=1}^n(c_j - c_j^\dagger)(c_{j+1} + c_{j+1}^\dagger) + \lambda c_jc_j^\...
-1
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1answer
154 views

Anti-commutative Hermitian operators in an infinite dimensional Hilbert space

An example of a pair of anti-commutative Hermitian operators in a finite Hilbert space is $\sigma_x$ with $\sigma_z.$ Indeed $\sigma_z\sigma_x=i\sigma_y$, whereas $\sigma_x\sigma_z=-i\sigma_y$. My ...
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0answers
319 views

Numerically Calculate expectation of $xp+px$?

I'm curious if there is a quick way to numerically calculate $\langle xp + px \rangle$ if we had the density function of our system. For example, if for x we can take the density function in the x-...
2
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2answers
106 views

Can the necessity of using anti-commutators for Dirac fields and commutator for Klein-Gorden be deduced from the field equations?

We all learned to use the commutator for quantizing the KG field and the anti-commutator for the Dirac field. We are told (which is correct) so that KG-excitations are bosons and Dirac-excitations ...
1
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2answers
11k views

Properties of anticommutators [closed]

Do anticommutators of operators has simple relations like commutators. For example: $$[AB,C]=A[B,C]-[C,A]B.$$ But I don't find any properties on anticommutators. Do same kind of relations exists ...
1
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2answers
197 views

Canonical quantization of bosons

During my studies on QFT a fundamental question occurred concerning the canonical quantization. In our course, we mentioned that: "The canonical quantization of a field with values in the complex ...
2
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1answer
2k views

Confusion about slash notation

I am confused about the slash notation and especially taking the square of a slashed operator. Defining $\displaystyle{\not} a \, = \, \gamma^\mu a_\mu$ we have $\,\,$ $\displaystyle{\not} a \...
0
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1answer
239 views

Fermion anti-commutation relations

The fermion anti-commutation relations are given as $$\{\psi_{\alpha}({\bf x},t),\psi_{\beta}^{\dagger}{(\bf x'},t)\} = \delta_{\alpha,\beta} \, \delta({\bf x} - {\bf x'}).$$ I am interested in ...
0
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1answer
77 views

Field operator anti-commutator relation

For the field operators (fermions) $$\hat{\Psi}^\dagger_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{-ikx}~\hat{a}^\dagger_{k,\sigma}$$ $$\hat{\Psi}_\sigma(x) = \dfrac{1}{\sqrt{V}}\sum_k e^{ikx}~\hat{a}...
2
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2answers
334 views

Second Quantization: Do fermion operators on different sites HAVE to anticommute?

In second quantization, we assume we have fermion operators $a_i$ which satisfy $\{a_i,a_j\}=0$, $\{a_i,a_j^\dagger\}=\delta_{ij}$, $\{a_i^\dagger,a_j^\dagger\}=0$. Another way to say this is that $$ ...
7
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2answers
856 views

Why do we use the anticommutation relation for particle-hole and chiral symmetries?

In physics we say that a quantity is conserved, if its operator commutes with Hamiltonian. For example, in condensed matter systems, when the momentum $k$ commutes with the Hamiltonian $H$ as $[H,k]=...
6
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1answer
476 views

Can I always find an unitary operator $B$ such that $\{A,B\}= 0$ for a given, unitary operator $A$?

Considering an arbitrary unitary operator $A$, what is the least criteria this operator must satisfy in order that it is possible to find at least another unitary operator $B$ that anti-commutes with ...
8
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4answers
2k views

Fermions, different species and (anti-)commutation rules

My question is straightforward: Do fermionic operators associated to different species commute or anticommute? Even if these operators have different quantum numbers? How can one prove this fact in a ...
6
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3answers
2k views

The anticommutator of $SU(N)$ generators

For the Hermitian and traceless generators $T^A$ of the fundamental representation of the $SU(N)$ algebra the anticommutator can be written as $$ \{T^A,T^{B}\} = \frac{1}{d}\delta^{AB}\cdot1\!\!1_{d} +...
2
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1answer
436 views

Derivation of fermion anticommutation rule

How one might derive the fermionic anticommutation relations? For bosonic particles, there is no ordering issue, and its commutation relation could be easily derived. However, for fermion, is there ...
3
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1answer
404 views

Minus sign in the time ordering operator

The time ordering operator is usually defined as $$\mathcal{T} \left\{A(\tau) B(\tau')\right\} := \begin{cases} A(\tau) B(\tau') & \text{if } \tau > \tau', \\ \pm B(\tau')A(\tau) & \text{if ...