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1answer
25 views

One stupid question about Dirac mass term and grassmannian numbers

Let's have Dirac mass term in lagrangian: $$ L_{M} = \bar{\Psi}\Psi $$ Lagrangian must be real-valued, i.e., its hermitian conjugation doesn't change it. But due to grassmannian nature of spinor ...
1
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0answers
34 views

Exponent rule for Grassmann numbers

I would like to ask if the following statement is legit for the Grassmann numbers $\eta^*,\xi_1,\xi_2,\xi_3$ and real parameters $A,B$ and $C$? $$e^{A \eta^*\xi_1} e^{B \eta^*\xi_2}e^{C \eta^*\xi_3} ...
0
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1answer
63 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 ...
4
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1answer
53 views

Non-hermiticity of Dirac Lagrangian: null momentum?

The usual Dirac Lagrangian is $L(\psi,\bar\psi)=\bar\psi(i\not\partial-m)\psi$. The canonical momenta are $$ \pi=\frac{\partial L}{\partial \psi_{,0}}=i\psi^\dagger \\ \bar \pi=\frac{\partial ...
1
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0answers
42 views

About the definition of super Hilbert Spaces

I have founded in the literature at leas two different definition of Hilbert spaces: Definition 1: A super Hilbert space is a complex super-vector space $\mathcal{H}=\mathcal{H}_0\oplus ...
0
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1answer
39 views

Polchinski, (0,0) picture vertex operator

I am currently working through chapter 12 of Polchinski and am confused as to how the equation $(12.3.39)$ for the (0,0) picture vertex operator arises. From the text: The state–operator mapping ...
1
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1answer
62 views

Do the Grassmann coordinates in the superfield formalism have any physical meaning?

In the superfield formalism we consider fields in a space who has four so called bosonic coordinates $x^{\nu}$ and four so called fermionic coordinates ...
3
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1answer
153 views

Wrong sign anticommutation relation for the Dirac field?

Consider the Dirac Lagrangian $$\mathcal{L}=\psi ^{\dagger }\gamma ^{0}\left( \mathrm{i}\gamma ^{\rho }\partial _{\rho }-m\right) \psi .$$ The conjugate momenta to $\psi ^{a}$ are defined, as usual, ...
2
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1answer
73 views

Why must the supersymmetry generators be spinors?

I have read in a few places (for example, at page 5 here) that the supersymmetry generators must be spinors. Quoting the reference mentioned The generator of the symmetry must relate two types of ...
1
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0answers
58 views

What mathematical structure describes superspace and superfields?

In every book related to supersymmetry I have encountered at some point the idea of superspace is introduced. Superspace is presented as a space spanned by 4 "normal" directions and 4 Grassmannian ...
3
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3answers
168 views

Is it possible to write the fermionic quantum harmonic oscillator using $P$ and $X$?

The Hamiltonian of the quantum harmonic oscillator is $$\mathcal{H}=\frac{P^2}{2m}+\frac{1}{2}m\omega^2X^2$$ and we can define creation and annihilation operators ...
5
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1answer
172 views

Basic question about superspace, Grassmann numbers and world sheet supersymmetry

So, I'm trying to read the section on superspace from the book on string theory by Becker, Becker and Schwarz, and I realized that I've been stuck on something simple for a while. Some relevant ...
1
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0answers
56 views

Notation - d.o.f.'s for Grassmann delta functions in a SUSY field theory amplitude

I was reading the following paper http://arxiv.org/pdf/1306.2962v1.pdf as I stumbled upon an issue concerning counting and assigning the Grassmann degrees of freedom that appear in grassmann delta ...
1
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2answers
62 views

Multivariable functions of Grassmann numbers

I'm trying to derive the closed form of the fermionic coherent state defined by the relation: $$ f_i|\vec{\eta}\rangle = \eta_i |\vec{\eta}\rangle \tag{4.10} $$ My book (Atland and Simons, Condensed ...
2
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1answer
95 views

Supersymmetry transformations as coordinate transformations

Usually, a supersymmetry transformation is carried out on bosonic and fermionic fields which are functions of the coordinates (or on a superfield which is a function of real and fermionic ...
2
votes
1answer
156 views

Treating the spinors as Grassmann numbers or as c-number objects

In the literature on supersymmetry, the following spinor summation convention is often used (eg. Wess & Bagger's book Supersymmetry and Supergravity) $$ \psi\chi = \psi^{\alpha}\chi_{\alpha} = ...
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2answers
104 views

Transposition of spinors

Suppose we have two 4-components Dirac spinors, that is two non commuting objects, $\psi$ and $\chi$. We know that: $ \bar{\psi} \chi= - \chi \bar{\psi} $ $ \bar{\psi} = \psi^{+} \gamma_0 $ $+=T*$ ...
3
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2answers
222 views

Why doesn't this multiplication of Grassmann variables give the expected result?

Would anyone be able to tell me how srednicki goes from step $(44.29)$ to $(44.30)$? Here is the paragraph: Now let us introduce the notion of complex Grassmann variables via $$\begin{align} ...
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0answers
39 views

Minimization of a quaradic-like expression when calculating the generating functional for free Dirac field

The generating functional for a free Dirac field is $$Z_0[\eta,\bar{\eta}]=\int D\bar{\psi}D\psi \mathrm{exp}\{i\int [\bar{\psi}(x)S^{-1}\psi(x)+\bar{\eta}(x)\psi(x)+\bar{\psi}(x)\eta(x)]dx\}$$ where ...
2
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0answers
106 views

Geometric interpretation of Grassmann variable

Grassmann variables were introduced to make path-integral formalism easier to handle fermionic (anti-commutating) fields. Mathematically they represent the exterior algebra of forms (or exterior ...
3
votes
2answers
271 views

Grassmann number representation for fermions

How one can simultaneously represent fermionic operators and its corresponding Grassmann variables, so that all the anticommutation relations between them and also states would take place? $$ ...
8
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2answers
794 views

Does the commutator of anything with itself not vanish?

In a quantum mechanics exam one question was to write the commutator of a couple of operators. Everybody got points taken away since they did not write $[Q_i, Q_i] = 0$ for all the operators $Q_i$ in ...
1
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1answer
226 views

Relative Minus signs from different Feynman Diagrams

I have a problem understanding the occurrence of a the relative minus signs between contributions, coming from different Feynman diagrams, involving fermions. A simple example is Bhabha scattering ...
2
votes
1answer
155 views

Lagrangian and grassmann numbers

Why sometimes we remember that "classical" lagrangians of fermions are constructed from grassmann numbers, while sometimes don't? For example, for Majorana's field in terms of 2-component spinors ...
2
votes
1answer
108 views

What happens to the free energy of the two-dimensional ising model with vortices?

The classical 2d Ising model has a Hamiltonian of the form: \begin{equation} H = -\sum_{m,n = 0}^{M,N} J_1 x_{m,n}x_{m+1,n} + J_2 x_{m,n}x_{m,n+1} \end{equation} The partition function for the model ...
4
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1answer
275 views

The correspondence between Grassmann number and 4-spinor

In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number. For two ...
0
votes
1answer
78 views

Fermion propagator is not a Grassmann-odd object?

Is the following differentiation correct: $$ \frac{\delta}{\delta\eta\left(z\right)}\int d^{4}yS_{F}\left(z-y\right)\eta\left(y\right) = S_F\left(z-z\right)$$ where $\eta$ is a Grassmann-valued ...
10
votes
1answer
249 views

Assumptions of the Coleman-Mandula Theorem

In the original paper All Possible Symmetries of the S-Matrix, by S. Coleman and J. Mandula, they prove their famous 'no go' theorem regarding the possible extensions of Poincaré symmetry. The ...
3
votes
1answer
342 views

Path integral as a functional determinant

In Peskin and Schroeder on pg. 304, the authors call the fermionic path integral: \begin{equation} \int {\cal D} \bar{\psi} {\cal D} \psi \exp \left[ i \int \,d^4x \bar{\psi} ( i \gamma_\mu D^\mu - m ...
1
vote
1answer
494 views

Gaussian Integral over Grassman variables

I need to evaluate two Grassmann integrals, one over "real" Grassmann variable another one over complex variables. Lets start with the real one first : The prototype we have for $n$ real Grassmann ...
4
votes
3answers
356 views

Completing the square for Grassmann variables

When working with path integrals of both bosonic and fermionic field variables, I'm a bit unsure of how to do the usual complete the square trick when an interaction between the two is concerned. Say ...
3
votes
1answer
684 views

Integrals over grassmann numbers

I want to prove an identity from Peskin&Schroeder, namely that $$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m^* \theta_l \exp(\theta_j^* B_{jk} \theta_k)=\det(B) ...
4
votes
1answer
487 views

Proof that eigenvalues of Fermionic creation/annihilation operators are Grassman numbers

It's stated probably in all textbooks on many-body functional integrals that operators that satisfy $$ \hat{a}^\dagger \hat{a} + \hat{a} \hat{a}^\dagger = 1 $$ must have eigenvalues that satisfy $$ ...
1
vote
1answer
137 views

Integration with Grassmann variables

How to show that $$ \int d\Psi d\bar {\Psi}e^{i \int d^{4}x\bar {\Psi} \hat {A} \Psi} = det (\hat {A})? $$ $\Psi , \bar {\Psi}$ refers to Dirac spinors (the second is $\bar {\Psi} = ...
2
votes
1answer
125 views

Product of Grassmann numbers

Let $\bar\theta$,$\theta$ be two Grassmann numbers. Then their product is a commuting number. If you were to integrate a function of the two $f(\bar\theta \theta)$ over $\bar\theta$,$\theta$ would it ...
1
vote
1answer
556 views

A few simple questions about Grassmann numbers: commutation relations and derivatives

I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions ...
2
votes
1answer
96 views

Determinant expansion

I've seen in a few textbooks now a useful looking expansion procedure for determinants, but I don't understand the details of it. Here is precisely the example I'm thinking of (Ex. 7.6). I don't ...
6
votes
1answer
275 views

Four Fermion Interactions

Given an action with a term like \begin{equation}S_{I}\sim \int\int (\psi^{\dagger}\psi)V(\psi^{\dagger}\psi)\end{equation} How do you evaluate this with a Fermionic path integral? I know the fields ...
1
vote
1answer
351 views

Grassmann fields according to Peskin and Schroeder

On page 301 in Peskin and Schroeder, they claim that a Grassman field $\psi(x)$ may be decomposed as $$\psi(x) = \sum_i c_i \phi_i(x),$$ where the $c_i$ are Grassmann numbers and the $\phi_i$ are ...
6
votes
1answer
170 views

Strange Grassmann double integration

I can unterstand why because the integration over Grassman variables has to be translational invariant too, one has $$ \int d\theta = 0 $$ and $$ \int d\theta \theta = 1 $$ but I dont see where ...
1
vote
0answers
168 views

Why are differential forms on a n-dimensional manifold a Grassmann algebra?

This is stated as an obvious example of a Grassmann algebra on page 32 in this tutorial I am trying to read, but to me it is unfortunately not so obvious. So can somebody expand this comment a bit ...
9
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1answer
494 views

What is the value of a quantum field?

As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
1
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3answers
193 views

Constructing Supersymmetric Lagrangians

It is a very trivial doubt but somehow I am not able to figure it out. While constructing a supersymmetric lagrangian we always even number of fermionic fields. One reason is of course the product ...
1
vote
1answer
259 views

dimensional analysis of Grassmann integration/differentiation

There is another paradox that I need to resolve: The Berezin integration rules for Grassmann odd variables give the same result as differentiation: If $f=x+\theta\psi$ is a superfunction, the ...
4
votes
1answer
285 views

Number of Grassmann generators for Dirac field?

How many Grassmann generators are sufficient for the description of a Dirac spinor in 4 dimensions? i.e. The Dirac field is a map to $\Lambda_N$, the space of supernumbers with $N$ real Grassmann ...
8
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3answers
1k views

Grassmann paradox weirdness

I'm running into an annoying problem I am unable to resolve, although a friend has given me some guidance as to how the resolution might come about. Hopefully someone on here knows the answer. It is ...
1
vote
1answer
150 views

Superspace Uncertainty Principle

Do the "operator for translations in superspace" and the "position in superspace operator" follow an uncertainty principle? How "real" is superspace? Aside from being weird (and possibly just a ...
1
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2answers
524 views

Nature of Derivatives of Anticommuting Variables

This may be a noob question but I've tried searching about this and haven't been able to put things into the context of what I've been studying. (Dot means the usual derivative w.r.t. time) If $c$ ...
5
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1answer
333 views

Grassmann Variables Representation?

It might be a silly question, but I was never mathematically introduced to the topic. Is there a representation for Grassmann Variables using real field. For example, gamma matrices have a ...
3
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2answers
551 views

A different type of Gaussian Grassmann Integral

Let $A$ be an antisymmetric matrix. Usually, one proves that for a Grassmann integral of the type, $$\int d\psi d\theta \exp( \psi^T A \theta) = \det(A)$$ where $\psi$ and $\theta$ are vectors of ...