[co-vector-of-Hermitian-tensor]
induces the co-vector of Hermitian-tensor
Since is a Hermitian matrix, then , so
Because is Hermitian or self-adjoint with respect to the inner product of , can be considered to act symmetrically on the two slots
The base of the vector space gives the coefficients of the co-vector
The action of on the co-vector is
It also corresponds to the transformation of the dual base, i.e., the base of the co-vector space , which is
parity dual and induced action
Similarly, co-vectors can also be defined for anti-Hermitian tensors
For complex conjugate two-tensors
[spinor-field-motivation]
-
formally corresponds momentum to gradient momentum , and to spacetime-momentum-spinor-representation
-
formally used for co-vector generated by to obtain field
-
action + product rule + divergence + zero boundary + integral quadratic form ==> self-adjoint operator
[massless-spinor-Lagrangian] alias [Weyl-Lagrangian]
or or
where is integrated using + quadratic form of as
The only one that works is , because is a divergence quantity, using Stokes' theorem + zero boundary
variation gives linear part
[massless-spinor-equation] alias [Weyl-equation]
or or
Similar to via , varying with respect to -valued is equivalent to varying with respect to -valued
can be interpreted as the gradient momentum of the field (after metric-dual) , compounded to, the multiplication of momentum and spinor
[Weyl-parity]
parity dual action uses spinor
parity dual eq
or or
[Weyl-eq-plane-wave]
Plane wave solution with and
A linear equation with indicates non-zero solutions. The solution space is one-dimensional, and the solution can be written as with
[massive-spinor-Lagrangian] action for mass-coupled spinors, alias [Dirac-Lagrangian]
couple Weyl spinors and their parity to
,
invariant non couple term
non-couple term variation with respect to gives
according to cancelation by parity
invariant couple term
couple term variation with respect to gives
- overall variation with respect to gives
- overall variation with respect to gives
- when , decouples into two parity-dual massless-spinors
These two PDEs imply
and as "square root of " [square-root-of-spacetime-Laplacian]
Overall , square root of KG. If a field satisfies the Dirac eq, then it satisfies the KG eq. Therefore, Dirac eq can approximate to KG eq, or further approximate to Schrodinger eq. But note that compared to the case of value, the difference from is that the angular momentum operator represented by or will have a spin part that affects the range space, similar to spinor-angular-momentum
All partial derivatives of the action is zero , giving [massive-spinor-equation] , alias [Dirac-equation]
Similar to via , variation with respect to valued is equivalent to variation with respect to valued
If the couple term is replaced by , the action is still invariant. However, the eq can no longer be decomposed into that simpler form
Question For any , the invariant matrix is probably only
[Dirac-eq-plane-wave]
Plane wave solution with and
The latter is a linear equation, so the solution is not difficult. The solution space is two-dimensional, and the solution can be written as (ref-17, p.100)
The conjugate phase plane wave satisfies the condition , and the solution can be written as
Similar to the scalar field case, superposition can be performed on the hyperboloid where the momentum lies. [linear-superposition-of-Dirac-eq]
[squrae-root-of-spacetime-momentum-spinor-representation]
Although it might be possible to use the eigenvalues of the Hermite matrix ( ), we will calculate it directly here. let or
==>
==> Use
==> Quadratic equation for : , solution
==>
or
Still Hermite. Calculation yields
Example
then
If , then
1,3 metric square root or square root . But you can also use to get the true square
Since , the transformation does not come from coordinate change
[motivation-of-gauge-field]
Ignored some issues
Tangent projective light cone bundle is well-defined
But is a field or a field needed? There are too many choices to lift a field to a field (or to a field); all lifting choices form a field
And there are only two ways to lift to
On a curved manifold, there may not even be a global single-valued lift.
The change in the lifting method from a field to a field corresponds to "changing the gauge", by multiplying the spinor by to change the gauge.
If the conserved current of the action is to be simpler, then use gauge transformation instead of . does not change the Lagrangian action, which simplifies the calculation of conserved currents (cf. the case of scalar field calculating 4-current for symmetry).
Changing the gauge is not compatible with taking derivatives of tangent spaces in bundle coordinates, so an additional structure โ connection โ must be introduced.
There are many possible connections. A good connection is one with the smallest curvature cf. electromagnetic-field
The bundle in curved spacetime can be directly defined in the bundle coordinates of the principal bundle (orthonormal frame bundle). Using the correspondence, changing bundle coordinates automatically corresponds to changing bundle coordinates.
In curved spacetime, one needs to deal with the covariant derivative of the spinor field with respect to the metric, which is derived from the metric-connection of the tangent vector field.
For spinors, one might need to use an orthonormal frame instead of a coordinate frame, i.e., use principal bundle. Does this introduce new difficulties for calculating covariant derivatives?
Even if the topology of the spacetime base manifold is non-trivial, there might exist different bundle types for gauge fields.
One problem is that, unlike spinor fields, gauge bundles do not seem to be directly related to tangent bundles.
It seems that all types of bundle types based on the base manifold must be considered simultaneously.
In the homotopy sense, has only one type of bundle type.
[spinor-field-gauge-imaginary-automorphism]
Although the cost is using dimensional spacetime, tangent space , but if we consider octonions , and if we choose a unit imaginary element to construct the spinor Lagrangian, then the octonion's imaginary automorphism group 's isotropy on leads to the decomposition , the isotropy group is isomorphic to , and it derives the action of on . cf. (ref-19, th.4)
Note that the octonion imaginary automorphism group is not . So this is not a group of gauge connection in the traditional sense.
It is said that is the gauge group of strong interaction. The advantage of the octonion method here is that there is no longer a need to additionally assume the separate existence of and and perform tensor operations on and Dirac spinor out of thin air. The tensor from comes from the connection of acting on the spinor or .
The isotropy in isomorphic to will change the phase or the gauge field of the gauge transformation, leading to the action on the part. Should it be said that there is a gauge field of the gauge transformation for the or gauge field of a gauge transformation, and then introduce the YangโMills eq of minimal curvature again like the electromagnetic field?
also cf. (ref-20, th.6)
Question What about the case of quaternions ?
[spin-connection]
The frame bundle of derived from the tangent bundle metric and the connection of the frame bundle derived from metric-connection behave as is locally type , acting on tangent vector fields by
The way to derive the spin-connection is to map the part of the induced metric-connection to in the orthonormal-frame, yielding the connection of the bundle, locally type , acting on the spinor field by with
Although the definition of the Pauli matrix for spin representation requires , both and Lie algebra can be expressed by the "square" of thereafter.
spin-connection also denoted by
[motivation-of-scalar-field] can scalar field be related to tautological bundle of projective-lightcone ?
According to the concept of spinor fields in spacetime manifold, "rotation by 720 degrees" and "parity" should occur in the tangent space construction, not in the spacetime manifold.
Since the tangent spaces of the spacetime manifold are all , can spinor fields be generalized to general spacetime manifolds?
[spinor-on-Lorentz-manifold] Question
massless-spinor-action
massless-spinor-equation
I haven't verified if this definition is conceptually reasonable. Compare with flat spacetime, try to prove or disprove it.
- is self-adjoint
- Only contributes to the variation of the action.
- i.e. square-root-of spacetime Laplacian (closer to the Laplacian of tangent vector fields rather than scalar fields)
massive-spinor-Lagrangian
massive-spinor-equation
Question As long as it locally quotients back from to , it can avoid the problem of continuous global single-valued lift to .
We know that the KG eq has a non-relativity approximation limit . Does the massive-spinor construction have a non-relativity approximation limit ?
Static doesn't need a non-relativity approximation limit , despite the presence of , just like static electromagnetic field equations don't need a non-relativity approximation limit . This is also true for the KG equation.
let static
static massless spinor eq
static massive spinor eq
They can couple to static electromagnetic gauge potential or just static electric or just static magnetic
In the presence of electromagnetic potential, the parity dual of massless particles might be different, for example, just static electric
When electromagnetic potential = 0, the parity dual eq is the same.