Questions tagged [superalgebra]

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One question about BRST symmetry in reading Srednicki’s book: Why should the BRST charge $Q_B$ be nilpotent?

In p.453, Srednicki claims that since the BRST transformation of a BRST transformation is zero, $Q_B$, the BRST charge, must be nilpotent: $$Q_{B}^{2}=0.\tag{74.32}$$ I don't know why.
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What's the space of eigenvalues/field configurations for a fermion?

In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\...
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1answer
62 views

Expanding superfields: inconsistency of notation?

If I have a wavefunction of a fermion field $\Psi[\psi]$ I can expand it like so about some vacuum: $$\Psi[\psi] = \Psi_0[\psi]( a + \int a(x)\psi(x)dx+\int a(x,y)\psi(x)\psi(y)dxdy+...)$$ Now all ...
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1answer
130 views

Representing quaternionic algebra with creation and annihilation operators?

The paper "Quantized Grassmann variables and unified theories" says given creation and annihilation operators $b$ and $b^\dagger$ one can represent quaternionic imaginary units $q_1$, $q_2$ and $q_3$ ...
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1answer
64 views

Why is the Jacobian factor for fermionic variables different from that for bosonic ones?

In Srednicki's textbook Quantum Field Theory, Section 77 discusses anomalies and the path integral for fermions. The path integral over the Dirac field is defined to be \begin{equation} Z(A) \equiv \...
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0answers
32 views

What does a supercharge physically conserve? [duplicate]

What is actually being conserved? I've calculated it for the Wess-Zumino model but I still have no idea what is actually being conserved due to Noether's Theorem. There is already a similar question, ...
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1answer
66 views

What is the definition of functions of Grassmann numbers?

I understand there are some relevant questions, but none of them solves my issue. From Atland and Simons (Condensed Matter Field Theory), the definition of functions of Grassmann numbers are defined ...
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2answers
87 views

Is the derivative with respect to a fermion field Grassmann-odd?

Fermion fields anticommute because they are Grassmann numbers, that is, \begin{equation} \psi \chi = - \chi \psi. \end{equation} I was wondering whether derivatives with respect to Grassmann numbers ...
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1answer
105 views

Why is supersymmetry a continuous symmetry?

Supersymmetry feels like a discrete symmetry to me, since the fermions are turning into bosons, and vice versa. I understand there is an infinitesimal parameter involved in the transformations, but I ...
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1answer
38 views

Stationary phase method for theories with both bosons and fermions

I am wondering if there exists a method to compute path integrals using the stationary phase method for theories with both bosons and fermions. (I am aware of such a method for theories with bosons ...
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1answer
70 views

What are Grassmann numbers in field theory?

I've been struggling with the use of Grassmann numbers in QFT e.g. Peskin and Schroeder. They are introduced as "numbers" whose product is antisymmetric, and associative (this isn't said, but used in ...
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1answer
134 views

Change of variables in path integral measure

In fermion's path integral we have a measure that you can write, in terms of the Grassmann variables $\psi, \bar{\psi}$ as $$ D\bar{\psi}D\psi, \quad \psi(x) = \sum_n a_n\phi_n(x), \quad \bar{\psi}(x)...
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1answer
63 views

Symmetry of the Batalin-Vilkovisky (BV) antibracket operation

Batalin and Vilkovisky define $^1$ an operation they call antibracket which is $$(F,H) = \Big(\frac{\partial_r F}{\partial \Phi^A}\Big) \Big(\frac{\partial_l H}{\partial \Phi^* _A} \Big) - \Big(\frac{...
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1answer
112 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 ...
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1answer
45 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} = -\...
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50 views

SUSY variation Wess-Zumino

I'm following John Terning book on Supersymmetry and in particular I'm trying to check the susy variations of the Wess-Zumino model given by $\mathcal{L}_s = \partial^\mu \phi^* \partial_\mu \psi \,...
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1answer
72 views

Intuition for the supertrace identity in supersymmetry

In pretty much every introductory book/lecture notes I've come across, one finds the expression for the mass matrices for scalars, fermions and vector bosons for a generic Lagrangian, and simply ...
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1answer
67 views

Transformation algebra of supersymmetry transformation

Given supersmmetry transformation: $$δX^\mu = i\barξψ^\mu.δψ^\mu = ξγ^a(∂_aX^\mu-\frac i2\barχ_a ψ^\mu). δχ_a = 2 D_aξ.$$ How to calculate the transformation algebra of $X^\mu$ and $\psi^\mu$? ...
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1answer
90 views

Can you quantize Grassmann-even superfields in the same fashion as Boson fields?

In a related Phys.SE question about supersymmetric Lagrangian $$ \mathcal{L} = - \frac{1}{2} (\partial S)^2 - \frac{1}{2} (\partial P)^2 - \frac{1}{2} \bar{\psi} \partial\!\!\!/ \psi, $$ the fields $S$...
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2answers
357 views

How are supersymmetry transformations even defined?

I am just starting to read about supersymmetry for the first time, and there is something bothering me. Supersymmetry transformations transform between bosonic fields and fermionic fields, but I don't ...
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2answers
162 views

Why do we want supersymmetry transformations to form a group?

I am getting introduced into supersymmetry reading Ryder "Quantum Field Theory". I have taken an introductory course on QFT last semester so I am far from being an expert. Sorry if this is a silly ...
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2answers
140 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 ...
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2answers
148 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 = ...
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1answer
137 views

Is there a type of supersymmetry where supercharges have spin 3/2?

Thinking of supersymmetry operators $Q$, they mix fields with a certain spin with fields with spin $1/2$ higher or lower. Thinking of open bosonic strings from string theory, the different modes are ...
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1answer
54 views

Could you get real space from Grassmann numbers?

You can get a vector field from a pair of spinor fields with $A_\mu(x)=\psi(x) \gamma_\mu \overline{\psi}(x)$. Using this fact could you define a space-time vector in terms of Grasman numbers? Say ...
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1answer
106 views

About the central charge of 4D extended supersymmetry algebra

The 4D SUSY algebra can be written as $$\{ Q_{\alpha}^{A} , Q_{\beta}^{B \dagger} \} = 2 m \delta^{AB} \delta_{\alpha \beta} + 2 i Z^{AB} \Gamma^0_{\alpha \beta}, \tag{B.2.37} $$ in a particular ...
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2answers
131 views

Cauchy-Schwarz inequality for Grassmann Integrals?

For square integrable functions $f,g$ of a real variable, the Cauchy-Schwarz inequality states that $$ \left(\int f(x)g(x)\,dx \right)^2 \le \int f(x)^2\,dx \int g(x)^2\,dx. $$ My question is: are ...
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1answer
111 views

Product of complex Grassmann numbers in higher dimensions

If two numbers $\eta$ and $\xi$ anti-commute. i.e., $$\eta\xi=-\xi\eta$$ they are called Grassmann numbers. It immediately follows that $$\eta^2=\xi^2=0,$$ and relations such as $$e^{a\eta}=1+a\eta;~~...
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380 views

An Identity for a Gaussian Grassmann integral from Wikipedia

I found this identity on Wikipedia: $$\int\exp\left[\theta^T A\eta+\theta^T J+K^T\eta\right]d\theta d\eta =\det A\exp\left[-K^TA^{-1}J\right],$$ where the integration variables are Grassmann ...
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1answer
292 views

How to construct a supersymmetry algebra?

Starting with the general notion of supersymmetry: $$Q| boson \rangle = | fermion \rangle \\ Q| fermion \rangle = | boson \rangle$$ I want to construct the superalgebra relations. After applying $...
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1answer
93 views

Can Grassmann-number variations of operators be represented by operators?

In my previous question, I asked about how to handle Grassmann-number variations of operators. I read a book that uses those variations $\delta \Phi = c \mathbb{1}$, with $c$ being a grassmann number ...
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48 views

Identifying superalgebras with fixed points under Cartan involution

I am making my way through the "Foundations of the $AdS_5 x S^5$ Superstring: Part I" paper by Arutyunov/Frolov 2009 (https://arxiv.org/abs/0901.4937v2) and am hoping someone can help me bridge a ...
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1answer
86 views

Extra Term in Super Particle Action

If you set up the first order formulation of a super particle $$S = \int dt \frac{1}{2}(\dot{x}^{\mu} \dot{x}_{\mu} - i \chi^{\mu} \dot{\chi}_{\mu})$$ by incorporating the constraints $$p^{\mu} p_{\...
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1answer
271 views

Localization and supersymmetry explanation

I'm confused about section 9.3 of the book Mirror Symmetry. In particular, I'm confused about the derivation carried out in equations (9.32) to (9.35) where they claim that the partition function is ...
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1answer
227 views

Two conjugations of supernumbers

Let $\theta$ and $\eta$ denote odd complex supernumbers (also known as Grassmann numbers), $a$ and $b$ arbitrary complex supernumbers. Say that $a$ is $\circ$-real (resp. $\circ$-imaginary) if $a^\...
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1answer
221 views

Eigenvalues of fermionic field operators

Consider the fermionic field operators $\psi_a(x), \psi^{\dagger}_b(y)$ with the canonical anti-commutation relations $$\{\psi_a(x),\psi_b(y)\} = 0 $$ and $$\{\psi^{\dagger}_b(t,\vec{x}),\psi_a(t,\vec{...
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2answers
315 views

What has “quantisation” to do with associated graded algebras?

I was currently reading an introduction into spin geometry by José Figueroa-O’Farrill. The first chapter handles Clifford algebras. When discussing the connection of the Clifford algebra to the ...
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323 views

Are Fock spaces just a special type of tensor algebra?

Are Fock spaces just a special type of tensor algebra? The definitions I am using: http://en.wikipedia.org/wiki/Fock_space http://en.wikipedia.org/wiki/Tensor_algebra From what I can tell, the ...
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107 views

The super Grassmannian $G_{2|2}(4|4)$

In the paper, the super Grassmannian $G_{2|2}(4|4)$ is defined by (12)--(18). An element of $G_{2|2}(4|4)$ can be written as a $(2 | 2) \times (4 | 4)$ matrix of full rank modulo the left action by ...
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1answer
76 views

Notation of supermatrices

I am trying to understand supermatrices. First I want to know the notation of supermatrices. In the paper, it is mentioned $(2|4|2) \times (2|2)$ supermatrices. What are $(2|4|2) \times (2|2)$ ...
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1answer
543 views

Number of supercharges in ${\cal N}\ge 1$ SUSY in $d\ge4$ dimensions

I have a couple of questions concerning supercharges and superalgebras: In four-dimensions, the minimal spinor representations are Weyl spinors. These have 4 real degrees of freedom (dof). This means ...
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119 views

Question about the expression of Witten Index

I am studying supersymmetry by myself. I do not understand the expression of Witten index, which is ${\rm Tr}(-1)^{F}$. What does it mean by writing $-1$ to the power of an operator $F$? Is this ...
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1answer
238 views

What are anticommuting spinor parameters $\zeta^\alpha$?

I'm reading Martin F.Sohnius, Introducing supersymmetry, page 82. It is the first time he introduces the anticommuting spinor parameters $\zeta^\alpha$ to calculate the supersymmetry variations of a ...
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0answers
51 views

Supersymmetric variation of $F$ term using super Jacobi identity [closed]

Basically, i am solving problems in note with a slightly different notation. \begin{align} [Q_\alpha F] = -i \lambda_\alpha (x), \quad [ \bar{Q}_{\dot{\alpha}},F] = -i \bar{\chi}_{\dot{\alpha}} \...
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2answers
148 views

Why $\delta F = B\epsilon$ and not $F=B \epsilon$ in supersymmetry?

We can express supersymmetric transformations as $$\delta F = B\epsilon, \tag{1} $$ $$\delta B = F\bar{\epsilon},\tag{2}$$ where $B$ and $F$ denote the bosons and fermions, respectively, in the theory ...
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1answer
79 views

Question about Supermatrix algebra

This question is inspired from a reading of Appendix F of P. van Nieuwenhuizen, Supergravity, Phys. Rep. 68 (1981) pp. 369-374. Consider a "supermatrix" $$M = \left(\begin{array}{cc} A & B\\ C &...
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53 views

N=4 d=3 susy algebra

Does anybody know how to derive the $\mathcal{N}=4$ d=3 susy algebra doing a dimensional reduction from the most famous $\mathcal{N}=4$ d=4? Equivalently, does it exist a reference in the literature ...
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1answer
297 views

Convert Grassmann numbers to real numbers [closed]

We know Grassmann numbers are complex numbers. Hence Grassmann integrals are also complex. How can we convert a Grassmann integral into real one, ie is there any transformation of converting complex ...
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1answer
231 views

Where does the “Supersymmetry” in Witten's proof of the Morse inequalities come from?

Where does the "Supersymmetry" in Witten's proof of the Morse inequalities (original paper and outline of proof for mathematicians) come from? Hopefully someone can provide an intuitive understanding? ...
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161 views

About the definition of super Hilbert Spaces

I have found in the literature at least two different definitions of $\bf{super}$ Hilbert spaces: Definition 1: A super Hilbert space is a complex super-vector space $\mathcal{H}=\mathcal{H}_0\oplus \...