Questions tagged [superalgebra]
131 questions
30
votes
3 answers
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 known that a superfunction (as a function of…
QuantumDot
- 6,531
23
votes
1 answer
Why/How is this Wick's theorem?
Let $\phi$ be a scalar field and then I see the following expression (1) for the square of the normal ordered version of $\phi^2(x)$.
\begin{align}
T(:\phi^2(x)::\phi^2(0):) &= 2 \langle 0|T(\phi(x)\phi(0))|0 \rangle^2 \\
&+ 4\langle…
user6818
- 4,749
13
votes
4 answers
Dirac equation as Hamiltonian system
Let us consider Dirac equation
$$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$
as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that Dirac equation becomes Hamiltonian equation…
Sasha
- 507
13
votes
1 answer
Basic Grassmann/Berezin Integral Question
Is there a reason why $\int\! d\theta~\theta = 1$ for a Grassmann integral? Books give arguments for $\int\! d\theta = 0$ which I can follow, but not for the former one.
F R
- 131
11
votes
1 answer
Integrals over Grassmann numbers
I want to prove an identity from Peskin&Schroeder, namely that
$$\left(\prod\limits_i^{} \int d \theta^*_i d\theta_i\right) \theta_m \theta_l^* \exp(\theta_j^* B_{jk} \theta_k)=\det(B) B^{-1}_{ml}\tag{9.70}$$
$B$ is a hermitean $N\times N$ matrix…
TheoPhysicae
- 397
9
votes
1 answer
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 $\hat\phi(\mathbf x)|\phi\rangle=\phi(\mathbf x)|\phi\rangle$, with field…
alexchandel
- 990
9
votes
3 answers
Embedding of $F(4)$ in $OSp(8|4)$?
Is the superconformal algebra in five dimensions, $F(4)$, a subalgebra of the (maximal) six-dimensional superconformal algebra $OSp(8|4)$?
Nikolay
- 91
9
votes
3 answers
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…
Yossarian
- 6,235
9
votes
2 answers
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 question. They said that they had to require this…
Martin Ueding
- 8,559
8
votes
1 answer
What are Grassmann (even/odd) numbers used in superalgebras?
Are Grassmann numbers a concept of graded Lie algebras or is something specific to superalgebras? What are they (i.e: how are they defined, important properties, etc.)? Is there a reasonable introduction to them?
I think that what makes me wonder a…
lurscher
- 14,933
8
votes
4 answers
How do you find a particular representation for Grassmann numbers?
This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the following representation:
Grassmann numbers can…
Physics_maths
- 5,717
8
votes
1 answer
What exactly are "Grassmann-valued fields"?
Peskin & Schroeder define a Grassmann field $\psi(x)$ as a function whose values are anticommuting numbers, that can be written as : [p.301 eq. 9.71]
$$\psi(x) = \sum\psi_i \phi_i(x),\tag{9.71}$$
where $\psi_i$ are said to be "grassmann numbers" and…
user341440
- 472
8
votes
1 answer
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 $a^\circ = a$ (resp. $a^\circ = -a$).
There are two…
Alex Shpilkin
- 1,122
- 9
- 25
8
votes
2 answers
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 exterior algebra, the author states:
Filtered algebras…
Peter Wildemann
- 298
7
votes
1 answer
What do the supercharges in extended supersymmetry do?
What do the supercharges in extended supersymmetry do?
In ${\cal N}=1$ supersymmetry there are a certain number of fermions and and equal number of bosons. You can transform all fermions to the bosons (and vice versa) in a 1 to 1 fashion using a…
Siraj R Khan
- 2,028