All Questions
Tagged with mathematical-physics spinors
21
questions
4
votes
1
answer
273
views
Physical motivation for the definition of Spin structure
I'm pretty confused about the motivations behind defining a spin structure on a manifold. Let me explain.
In quantum mechanics, particles are represented by irreducible unitary projective ...
- 123
1
vote
1
answer
196
views
Why the Double Covering?
It is known mathematically that given a bilinear form $Q$ with signature $(p,q)$ then the group $Spin(p,q)$ is the double cover of the group $SO(p,q)$ associated to $Q$, and that $Pin(p,q)$ is the ...
- 1,611
0
votes
1
answer
114
views
Why the symmetry is not $Pin(1,3)$ or $Pin(3,1)$ in condensed matter physics?
In usual electron systems (or condensed matter physics), it is well known that $T^2=-1$ and $M^2=-1$, where $T$ and $M$ are time reversal and reflection along some axis. But in general, the symmetry ...
- 51
5
votes
1
answer
228
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How to relate mathematicaly rigorous spinor fields to the ones used in physics?
One way to rigorously define spinor fields on metric manifolds is through the language of associated bundles. Namely, we have a principal bundle $P \overset{\pi}{\longrightarrow} M$ over $\mathrm{Spin}...
- 63
5
votes
1
answer
242
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In what sense does a pure spinor represent the orientation of a unique spacelike codimension-2 plane?
References 1 and 2 define a pure spinor $\psi$ to be a solution of the Cartan-Penrose equation
$$
\newcommand{\opsi}{{\overline\psi}}
v^\mu\gamma_\mu\psi=0
\hspace{1cm}
\text{with}
\hspace{1cm}
v^\mu\...
- 54.5k
1
vote
0
answers
153
views
Lie algebra generators as rank-16 matrix spinor representations of $ππππ(10)$
A simple Lie group $ππππ(10)$ has a spinor representations of 16 dimensions, which is distinct from the vector representation of 10 dimensions (coming from standard vector representation of SO(10))...
2
votes
1
answer
271
views
Spin structures and boundary conditions for worldsheet fermions
The definition I'm aware of a spin structure is the following one:
Definition: Let $(M,g)$ be a semi-Riemannian manifold with signature $(p,q)$. Let ${\cal F}M$ be the principal ${\rm SO}(p,q)$-...
- 36.4k
1
vote
0
answers
177
views
Majorana fermions in Euclidean and Minkowski signatures - contradiction with Wikipedia Table
In this wonderful lecture note on Clifford Algebra and
Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf
Somehow I find some inconsistency with his Tables of Euclidean and ...
- 4,336
2
votes
1
answer
84
views
The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions
Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions)
My question is that what does the conjugacy mean ...
- 4,336
1
vote
1
answer
266
views
How these two approaches to spinors in curved spacetimes relate?
Regarding spinors in curved spacetimes I have seem basically two approaches. In a set of lecture notes by a Physicist at my department he works with spinors in a curved spacetime $(M,g)$ by picking a ...
- 36.4k
1
vote
0
answers
123
views
How to know if a spinor $\Psi\left(x\right)$ is the ground state of the system?
Suppose we have time independent one-dimensional single particle SchrΓΆdinger-like equation$$-\frac{d}{dx}\left(A\left(x\right)\frac{d}{dx}\psi\left(x\right)\right)+V\left(x\right)\psi\left(x\right)=E\...
- 11
1
vote
1
answer
51
views
Suppose I give you $2^N$ functions that are eigenvectors of a fermionic $H$. How do I determine which function describes which spin configuration?
Consider the hamiltonian
$$
H = - \frac{1}{2} \nabla^2 + V.
$$
The potential $V : (\mathbb{R^3})^N \to \mathbb{R}$ is symmetric, so for each eigenvalue, there is an antisymmetric eigenvector. There is ...
- 1,356
1
vote
0
answers
169
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$\operatorname{GL}(4, R)$ group and Spinors in general relativity
Spinors are special representations of $\operatorname{Spin}(n)$ group which is double cover of $\operatorname{SO}(n)$. I am familiar with tetrad formalism and spin connection. $\operatorname{GL}(n,R)$ ...
- 463
2
votes
0
answers
93
views
Motivating the Unintuitive Properties of Spinors
In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
- 130
2
votes
0
answers
82
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Spinors, punctured plane and principle frame bundle
I am reading Applied Conformal Field Theory by Ginsparg. On page 72, while describing different boundary conditions on fermion he states the following.
We shall choose to consider periodic $(P)$ ...
- 1,542