What is the official mathematical concept behind the principle of the anti commutator made use of to quantize fermions?

I recognize Lie groups are specified by the framework constants related to the lie brackets, which are dealt with as commutators in quantum technicians, yet i do not recognize of a mathematics concept connected to team concept to specify or make use of an anti commutator. If Lie groups concept makes use of the commutator, what concept makes use of the anti commutator?

Limited teams (not Lie groups, which are continual), can be defined by framework formulas similar to a Lie brace, yet extra basic, of which commutation or anti commutation relationships are simply among boundless opportunities. Exists such selection of feasible framework formulas in continual teams also?

2019-12-02 02:51:34
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Answers: 2

The algebraic framework matching most normally to the anticommutator is that of the Clifford algebras.

One can construct a Clifford algebra by specifying $n$ creating components $\mathbf{e}_j$ with $j \in \{1,\ldots,n\}$ such that they please the problems :

$$ \mathbf{e}_i \mathbf{e}_j + \mathbf{e}_j \mathbf{e}_i = 2 \delta_{ij} \; ,$$

where $\delta_{ij}$ is the Kronecker delta. From these components you can create items of components of the kind

$$\mathbf{e}_{i_1}\mathbf{e}_{i_2}\ldots\mathbf{e}_{i_p} \; ,$$

for $p$ differing from $0$ to $n$. There is a total amount of $2^n$ items that can be created and also from these, taking straight mix you can create the remainder of the algebra.

You can examine that Pauli matrices and also quaternions create Clifford algebras.

Currently, if you take a Clifford algebra with $n$ creating components, and also construct the components $M_{ij}=\frac{1}{2}\mathbf{e}_{i}\mathbf{e}_{j}$, you can show that they create the basis of the Lie algebra $so(n)$.

An additional construct pertaining to fermions, and also which is necessary in the course indispensable depiction of fermions, are the Grassmann numbers.

2019-12-03 05:17:18

If Lie algebras are (taking into account the Poincare - Birkhoff - Witt theory) a full axiomatization of the antisymmetric reproduction $AB-BA$ in an associative algebra, Jordan algebras are a virtually - full axiomatization of the symmetrical reproduction $(AB+BA)/2$. (http://en.wikipedia.org/wiki/Jordan_algebra ). They are the closest point recognized to an inherent framework pertaining to anticommutators.

Unlike Lie algebras, there are phenomenal Jordan algebras that do not originate from the reproduction in associative algebras, and also Jordan algebras are not recognized to be infinitesimal things for teams or any kind of various other framework. So they do not emerge as usually beyond (or in) quantum technicians and also the first wish for a Lie - like concept have actually not been understood.

If you suggest not the anti - commutator yet the supercommutator, $AB \pm BA$ with the indicator relying on parity of $A$ and also $B$ or their consituents (as an example, adverse for bosons and also favorable for fermions) there is a concept of Lie superalgebras and also Lie supergroups. There are new sensations in the category, such as continual family members of nonisomorphic straightforward things. The concept can be phrased as the "Lie team concept in the group of extremely - vector spaces", which subsequently is a grandfather clause of Lie concept (or team concept, or algebraic geometry) in a tensor group. So in concept there are several feasible concepts of commutator - like things, yet I do not recognize if any kind of have actually been located to be intriguing besides the common concept, its extremely - variation, and also analogues in particular $p$.

2019-12-03 04:18:12