Questions tagged [grassmann-numbers]
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275
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Weinberg's path integral for fermions in Volume 1
In sec. 9.5 of Weinberg's QFT 1, he introduces operators $Q$ and $P$ satisfying
$$
\{Q,P\}=i \tag{9.5.1}
$$
$$
\{Q,Q\}=\{P,P\}=0 \tag{9.5.2}
$$
and eigenstates $|q\rangle$:
$$
Q|q\rangle =q|q\rangle \...
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When does the spinor need to be in a Grassmann variable?
Follow the closed question When does the spinor need to be in a grassmann variable?
--
Does the spinor in the spinor representation of the space-time symmetry
Lorentz space-time symmetry, like $so(1,...
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Calculating a Gaussian-like path integrals with Grassmann variables and real variables
I want to compute the following path integral
$$Z[w] = \frac{1}{(2\pi)^{n/2}}\int d^n x \: \prod_{i=1}^{n}d\overline{\theta}_id\theta \: \exp{\left(-\overline{\theta}_i \partial_j w_i(x)\theta_j -\...
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2
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Grassmann parameter in supersymmetry
Let's consider a free Wess-Zumino Lagrangian given by
$$\mathcal{L} = \partial^{\mu}\overline{\phi}\partial_{\mu}\phi + i\psi^{\dagger}\overline{\sigma}^{\mu}\partial_{\mu}\psi\tag{1}$$
Whose ...
user366054
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1
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Bosonic representation of delta function for Grassmann-even quantity
Suppose I have 2 Grassmann scalars $\theta$ and $\bar{\theta}$ and form the bosonic quantity $X = \bar{\theta}\theta$. Is there a purely bosonic representation of the delta function $\delta(X - \...
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Spin of covariant derivatives in 3D $\mathcal{N}=2$ superspace
Consider the 3D $\mathcal{N}=2$ superspace covariant derivatives $D_\alpha$ and $\bar D_\beta$, which have the following anti-commutation relations
$$\{ D_\alpha, \bar D_\beta \}=-2i \gamma^\mu_{\...
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What are self-interacting fermions?
There're a bunch of models of fermions with quartic self-interactions. There's an introduction from this wikipedia page.
For example, one can construct the Soler model of self-interacting Dirac ...
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Gaussian integral identity over Grassmann numbers
I am reading Zee's Quantum Field Theory in a Nutshell. In the section about Grassmann numbers, there is an identity:$$\int dx\int dy\,e^{yAx}=\det A\tag{II.5.13}$$ where $x=(x_1,x_2,\dots,x_N),y=(y_1,...
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Why can the commutator of a general expression be replaced by the anti-commutator in the $bc$ CFT theory?
Polchinski states in his equation 2.6.14 (in his book String Theory Vol. 1, Introduction to the Bosonic String) that for charges $Q_1$ and $Q_2$ the following equation holds, where $j_i$ is the ...
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Grassmann numbers for fermions in QFT
I'm studying the Grassmann variables from Polchinski's string theory textbook appendix A. On page 342,
In order to follow the bosonic discussion as closely as possible, it is useful to define states ...
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0
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20
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Symmetries and equations of superspheres and other superspaces
What are the symmetries and the most studied/most standard examples of superspaces? I include exceptional superalgebras and infinite-dimensional spaces.
Bonus: Do quantum groups apply in the above ...
- 6,165
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votes
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answer
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Contour integral for commutator of fermionic fields
Suppose we have primary fields $A$ and $B$ which have the OPE,
$$A(z) B(w) = \frac{1}{z-w} = -B(z)A(w), \quad |z| > |w|,\tag{1}$$
so they have fermionic statistics. Now I was curious how this would ...
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Gaussian integral over Grassmann numbers
I'm trying to evaluate a Gaussian integral over Grassmann numbers but not sure if I've made a mistake.
What I want to evaluate is
\begin{equation}
\left(\prod^N_i\int d\theta^*_i d\theta_i\right)\...
- 11
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2
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Fermionic measure in path integral
When writing the fermionic path integral one arrives at an expression containing $\mathcal{D}\bar{\psi}$ and $\mathcal{D}\psi$:
$$
\int \mathcal{D}\bar{\psi} \mathcal{D}\psi e^{iS}
$$
Usual ...
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Matrix representation of Grassmann variables and Berezin Integrals
In this question, the problem of finding matrix representations for a set of Grassmann variables is discussed.
How can this representation be used in Berezin integrals or Grassmann derivatives?
Can ...
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Grassmann numbers
I am reading Zee's Quantum Field Theory in a Nutshell and am having some questions about Grassmann numbers. Let $x,y$ be Grassmann numbers. I think I have two relations:
$$e^{x+y}=1+x+y\tag{1}$$ and
$$...
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About the Hilbert space that carries the representation of $\{\psi (x), \bar{\psi (y) }\}=i\delta (x-y) $?
What is this Hilbert space? Is it the complex Hilbert space of wavefunctionals spanned by using the spinor-field configurations as the basis vectors?
I know that the wavefunctional space carries a ...
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Jacobian functional matrix for fermionic path integral
I am revisiting Srednicki's book Chapter 77 and struggling to understand how you define the change of variables in the fermionic field integral
Srednicki defines the Jacobian functional matrix for the ...
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Spinor field path quantization
Although I have asked a similar question here, here, I find that I don't totally understand it, so I arrange my new ideas to this post.
Begin with Berezin integral:
$$\left(\prod_i \int d \theta_i^* d ...
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Functions on Superspace
I find the definition of a function $\phi$ on a superspace $(z,\theta)$ confusing for the following reason. $\phi(z,\theta)$ can be expanded as $$\phi(z,\theta) = \phi_0(z) + \theta \psi(z)\tag{1}.$$ ...
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Is this interpretation of fermionic dimensions correct?
The number of Grassmann coordinates in ${\cal N}=1$, $3+1$ dimensional superspace is $4$. Let's call them:
$\theta_1$ $\theta_2$ $\theta_3$ $\theta_4$.
The Grassmann variables can be represented by ...
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votes
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Spinor functional quantization unitarily equivalent and determinant
On P&S's qft page 301 and 302, the book discussed functional quantization of spinor field.
The book define a Grassmann field $\psi(x)$ in terms of any set of orthonormal basis functions:
\begin{...
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How do you show that $DD \Phi = 4 m \Phi^{\dagger}$ yields massive field equations for the component fields of a chiral superfield $\Phi$?
Suppose $\Phi(x, \theta, \bar{\theta})$ is a chiral superfield ($\bar{D}_{\dot{\alpha}} \Phi = 0$). One can write it in components as
$$
\Phi(x, \theta, \bar{\theta}) = f(x) + \theta \phi(x) + \bar{\...
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An identity for Grassmann Gaussian integrals
Suppose, for each $i=1,...,N$, $\bar{\psi}_{i}$ and $\psi_{i}$ are Grassmann vectors with $n$ entries $\bar{\psi}_{i,1},...,\bar{\psi}_{i,n}$ and analogously for $\psi_{i}$. Let $d\mu_{C}(\bar{\psi},\...
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Grassmann numbers and fermionic strings
Is it correct that by introducing Grassmann numbers as new directions of spacetime we can make strings behave like fermions (that is, 1/2-spin objects)?
And if so, is it possible to show how that ...
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Contradictory results for Berezin integral
Say $$u = (u_1, \dots, u_{2n}) = (\xi_1, \eta_1, \dots \xi_n, \eta_n)\tag{1}$$ is a vector of Grassmann variables. For an antisymmetric bosonic matrix $A$ we know that
$$
\int e^{\frac{1}{2} \sum_{a,...
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Complex Grassmann Dirac Functional - How do we integrate over it?
I'm following the Book of Brian Hatfield (Quantum Field Theory of point particles and Strings), p.192 here: For real Grassmann numbers (and Functionals thereof):
If $\Phi[\psi]$ is a functional, and $\...
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How do I differentiate a spinor?
I was trying to derive the Euler-Lagrange equation for a lagrangian and this isn't right.
\begin{align}\frac{\partial \epsilon ^{\dot{a}\dot{b}} \psi^\dagger _\dot{a} \psi^\dagger _\dot{b}}{\partial \...
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By using a Hilbert space (enhanced by Grassmann Numbers), can we write down a full set of eigenstates of the fermionic field operator?
By extending the Hilbert space, using grassmann numbers instead of complex numbers, we can write down eigenstates of the fermionic annihilation operator $a$ without getting into trouble with the ...
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Motivation of Grassmann fields in the Faddeev-Popov method for free Gluon fields
The Faddeev-Popov approach to make the generating functional corresponding to free gluon fields well defined, introduces two independent Grassmann fields. Since these are scalar, their quanta can be ...
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Grassmann property of Fermion field in Hilbert space and spinor space [duplicate]
If we consider the trace of two fermion field ($\psi_A \ \psi_B$), and using the cyclic property of trace (which has Grassmann property in general I thought).
$$
\text{Tr}(\psi_A \ \psi_B)=(-)\text{Tr}...
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votes
2
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Connection between column matrix and Grassmann numbers in Dirac field
In canonical quantization the Dirac equation is a complex column matrix, while in path integral formulation it's Grassmann numbers.
Is there a formula to convert from complex matrix to Grassmann ...
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What is the problem with classical fermionic field?
Consider classical fermionic field. We have it's action, equations of motion and so we can get it's solutions, right?
For example, we can consider gravitational solutions with fermions (in particular, ...
user341530
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2
answers
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Application of "real" Grassmann Gaussian integrals
In Appendix 2B of the CFT yellow book by Francesco et al, the authors introduced two types of Grassmann Gaussian integrals (the $\theta$'s below are generators of a Grassmann algebra):
The "real&...
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Born's Rule for states over supernumbers?
For Quantum-mechanics on a Hilbert-space over the complex numbers, the usual scalar product of two states $\langle \phi | \psi \rangle$ and gives the transition amplitude between the two states. The ...
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votes
2
answers
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views
Gaussian Grassmann integral with complex/bosonic source term
I'm interested in solving the following multi-dimensional integral
$$
\int d \theta d \bar{\theta} e^{-\bar{\theta}M \theta +\Lambda \theta + \bar{\theta} J }
$$
where $\theta$ is a $N$-dimensional ...
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How does the Pauli Exclusion Principle work in Twistor theory?
Twistor theory is described sometimes as a 'natural' way to represent spinor fields.
In QED, we have the Grassmann valued spinor field $\Psi^\alpha(x)$, which naturally leads to the exclusion ...
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Is it wrong to say that nature is in a superposition of fermionic coherent states?
We can't observe grassmann numbers in nature, and physical systems "in nature" are never in a fermionic coherent state (whose eigenvalues are grassmann numbers).
However, if for example in ...
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votes
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Are Grassmann numbers always "under the hood" if we deal with fermionic ladder / field operators?
For the set of all fermionic field operators $\Psi(x) | x \in \mathbb{R}^{3 +1}$, we won't find a $|\phi \rangle$ that is an eigenstate to the complete set of field operators, unless we make use of ...
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What Object is the Dirac Lagrangian in the functional treatment of QFT, where $\Psi$ and $\bar{\Psi}$ are Grassmann-numbers?
As far as I understood, in the path integral formulation of QFT, a field configuration is modelled by a mapping
$$
x \rightarrow \Psi(x)
$$
Where $\Psi(x)$ are 4 components, each represented by 4 ...
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votes
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Does this Fermion path integral have a solution?
This is a question on the mathematics of path integration.
If we take the action density $S[\phi](t) = \frac{1}{2}\dot{\phi}\dot{\phi}$ and we take the path integral
$$K_T(A,B) = \left. \int e^{-\...
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Path integral on many-body quantum mechanics
Suppose $\mathscr{H}$ is a Hilbert space describing a one-particle quantum system and $\mathcal{F}(\mathscr{H})$ is its associated Fock space, which is used to describe a many-body quantum system. Let ...
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Definition of a determinant Peskin&Schroeder
In page 514 of Peskin&Schroeder we are given the definition of a determinant as
$$
\det\left(\frac{1}{g}\partial_\mu D^\mu\right)=\int{\cal{D}cD\bar{c}\exp\left[i\int {d^4x\bar{c}(-\partial^\mu D_\...
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What does it exactly mean by right and left functional derivatives?
In BV formalism of the gauge theory, we need to compute the right / left functional derivatives of the actions that include fermions.
I do not quite see what it means by that.
For example, let us ...
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Anti-Symmetry of Dirac Operator
In his paper Fermion Path Integrals And Topological Phases, Witten states
“Whenever one has a theory of fermions, the quadratic part of the fermion action is always antisymmetric by virtue of fermi ...
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Feynman rules in Grassmann variables [duplicate]
I'm given the following integral $$ Z[w] = \frac{1}{ (2 \pi)^{n/2}} \int d^n x \prod_{i=1}^n d \overline{\theta}_i d \theta_i \exp \left( - \overline{\theta}_i \partial_j w_i (x) \theta_j - \frac{1}{2}...
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Complex valued Grassmann variables $(\theta \eta)^* $, $(\theta \eta)^T$ and $(\theta \eta)^\dagger$
Since hermitian conjugation and complex conjugation are serious issues in a QFT lagrangian with Grassmann variables, see here and here. Let us try to go to the bottom.
We start by accepting the ...
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Path integral for complex scalar field
I am taking a QFT course which focuses on the path integral formulation. At a certain point, I was confused because we saw that, when integrating over complex Grassmann fields for fermions, we defined ...
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Poisson bracket to quantum commutator for Grassmann-valued coordinates
In Dirac's Principles of Quantum Mechanics (pg. 86 eq. 5), the quantum commutator is motivated by looking for a bracket that satisfies the same properties of the Poisson bracket. When deriving the ...
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How do we trace over subregions in a fermionic QFT?
Bosonic Case
In a bosonic QFT, the Hilbert space associated to a surface $\Sigma$ is the appropriate space of wavefunctionals on $\Sigma$. Hence, if $\Sigma=\Sigma_1 \sqcup \Sigma_2$, we find that the ...
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