In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
Dirac spinors are important and interesting in numerous ways. Foremost, they are important as they do describe all of the known fundamental particle fermions in nature; this includes the electron and the quarks. Algebraically they behave, in a certain sense, as the "square root" of a vector. This is not readily apparent from direct examination, but it has slowly become clear over the last 60 years that spinorial representations are fundamental to geometry. For example, effectively all Riemannian manifolds can have spinors and spin connections built upon them, via the Clifford algebra.[1] The Dirac spinor is specific to that of Minkowski spacetime and Lorentz transformations; the general case is quite similar.
This article is devoted to the Dirac spinor in the Dirac representation. This corresponds to a specific representation of the gamma matrices, and is best suited for demonstrating the positive and negative energy solutions of the Dirac equation. There are other representations, most notably the chiral representation, which is better suited for demonstrating the chiral symmetry of the solutions to the Dirac equation. The chiral spinors may be written as linear combinations of the Dirac spinors presented below; thus, nothing is lost or gained, other than a change in perspective with regards to the discrete symmetries of the solutions.
The remainder of this article is laid out in a pedagogical fashion, using notations and conventions specific to the standard presentation of the Dirac spinor in textbooks on quantum field theory. It focuses primarily on the algebra of the plane-wave solutions. The manner in which the Dirac spinor transforms under the action of the Lorentz group is discussed in the article on bispinors.
Definition
editThe Dirac spinor is the bispinor in the plane-wave ansatz
An explanation of terms appearing in the ansatz is given below.
- The Dirac field is , a relativistic spin-1/2 field, or concretely a function on Minkowski space valued in , a four-component complex vector function.
- The Dirac spinor related to a plane-wave with wave-vector is , a vector which is constant with respect to position in spacetime but dependent on momentum .
- The inner product on Minkowski space for vectors and is .
- The four-momentum of a plane wave is where is arbitrary,
- In a given inertial frame of reference, the coordinates are . These coordinates parametrize Minkowski space. In this article, when appears in an argument, the index is sometimes omitted.
The Dirac spinor for the positive-frequency solution can be written as
- is an arbitrary two-spinor, concretely a vector.
- is the Pauli vector,
- is the positive square root . For this article, the subscript is sometimes omitted and the energy simply written .
In natural units, when m2 is added to p2 or when m is added to , m means mc in ordinary units; when m is added to E, m means mc2 in ordinary units. When m is added to or to it means (which is called the inverse reduced Compton wavelength) in ordinary units.
Derivation from Dirac equation
editThe Dirac equation has the form
In order to derive an expression for the four-spinor ω, the matrices α and β must be given in concrete form. The precise form that they take is representation-dependent. For the entirety of this article, the Dirac representation is used. In this representation, the matrices are
These two 4×4 matrices are related to the Dirac gamma matrices. Note that 0 and I are 2×2 matrices here.
The next step is to look for solutions of the form
Results
editUsing all of the above information to plug into the Dirac equation results in
Solve the 2nd equation for χ and one obtains
Note that this solution needs to have in order for the solution to be valid in a frame where the particle has .
Derivation of the sign of the energy in this case. We consider the potentially problematic term .
- If , clearly as .
- On the other hand, let , with a unit vector, and let .
Hence the negative solution clearly has to be omitted, and . End derivation.
Assembling these pieces, the full positive energy solution is conventionally written as
Solving instead the 1st equation for a different set of solutions are found:
In this case, one needs to enforce that for this solution to be valid in a frame where the particle has . The proof follows analogously to the previous case. This is the so-called negative energy solution. It can sometimes become confusing to carry around an explicitly negative energy, and so it is conventional to flip the sign on both the energy and the momentum, and to write this as
In further development, the -type solutions are referred to as the particle solutions, describing a positive-mass spin-1/2 particle carrying positive energy, and the -type solutions are referred to as the antiparticle solutions, again describing a positive-mass spin-1/2 particle, again carrying positive energy. In the laboratory frame, both are considered to have positive mass and positive energy, although they are still very much dual to each other, with the flipped sign on the antiparticle plane-wave suggesting that it is "travelling backwards in time". The interpretation of "backwards-time" is a bit subjective and imprecise, amounting to hand-waving when one's only evidence are these solutions. It does gain stronger evidence when considering the quantized Dirac field. A more precise meaning for these two sets of solutions being "opposite to each other" is given in the section on charge conjugation, below.
Chiral basis
editIn the chiral representation for , the solution space is parametrised by a vector , with Dirac spinor solution
Spin orientation
editTwo-spinors
editIn the Dirac representation, the most convenient definitions for the two-spinors are:
Pauli matrices
editThe Pauli matrices are
Using these, one obtains what is sometimes called the Pauli vector:
Orthogonality
editThe Dirac spinors provide a complete and orthogonal set of solutions to the Dirac equation.[2][3] This is most easily demonstrated by writing the spinors in the rest frame, where this becomes obvious, and then boosting to an arbitrary Lorentz coordinate frame. In the rest frame, where the three-momentum vanishes: one may define four spinors
Introducing the Feynman slash notation
the boosted spinors can be written as
The conjugate spinors are defined as which may be shown to solve the conjugate Dirac equation
with the derivative understood to be acting towards the left. The conjugate spinors are then
The normalization chosen here is such that the scalar invariant really is invariant in all Lorentz frames. Specifically, this means
Completeness
editThe four rest-frame spinors indicate that there are four distinct, real, linearly independent solutions to the Dirac equation. That they are indeed solutions can be made clear by observing that, when written in momentum space, the Dirac equation has the form
This follows because
With an appropriate choice of the gamma matrices, it is possible to write the Dirac equation in a purely real form, having only real solutions: this is the Majorana equation. However, it has only two linearly independent solutions. These solutions do not couple to electromagnetism; they describe a massive, electrically neutral spin-1/2 particle. Apparently, coupling to electromagnetism doubles the number of solutions. But of course, this makes sense: coupling to electromagnetism requires taking a real field, and making it complex. With some effort, the Dirac equation can be interpreted as the "complexified" Majorana equation. This is most easily demonstrated in a generic geometrical setting, outside the scope of this article.
Energy eigenstate projection matrices
editIt is conventional to define a pair of projection matrices and , that project out the positive and negative energy eigenstates. Given a fixed Lorentz coordinate frame (i.e. a fixed momentum), these are
These are a pair of 4×4 matrices. They sum to the identity matrix:
It is convenient to notice their trace:
Note that the trace, and the orthonormality properties hold independent of the Lorentz frame; these are Lorentz covariants.
Charge conjugation
editCharge conjugation transforms the positive-energy spinor into the negative-energy spinor. Charge conjugation is a mapping (an involution) having the explicit form
See also
editReferences
edit- ^ Jost, Jürgen (2002). "Riemannian Manifolds". Riemannian Geometry and Geometric Analysis (3rd ed.). Springer. pp. 1–39. doi:10.1007/978-3-642-21298-7_1. See section 1.8.
- ^ Bjorken, James D.; Drell, Sidney D. (1964). Relativistic Quantum Mechanics. McGraw-Hill. See Chapter 3.
- ^ Itzykson, Claude; Zuber, Jean-Bernard (1980). Quantum Field Theory. McGraw-Hill. ISBN 0-07-032071-3. See Chapter 2.
- Aitchison, I.J.R.; A.J.G. Hey (September 2002). Gauge Theories in Particle Physics (3rd ed.). Institute of Physics Publishing. ISBN 0-7503-0864-8.
- Miller, David (2008). "Relativistic Quantum Mechanics (RQM)" (PDF). pp. 26–37. Archived from the original (PDF) on 2020-12-19. Retrieved 2009-12-03.