Complex four-component spinor
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
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The
Dirac spinor
is the
bispinor
in the
plane-wave
ansatz
of the free
Dirac equation
for a
spinor
with mass
,
which, in
natural units
becomes
and with
Feynman slash notation
may be written
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
where
- 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
m
2
is added to
p
2
or when
m
is added to
,
m
means
mc
in ordinary units; when
m
is added to
E
,
m
means
mc
2
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
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The 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
while at the same time splitting
ω
into two two-spinors:
Results
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Using all of the above information to plug into the Dirac equation results in
This matrix equation is really two coupled equations:
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
The above introduces a normalization factor
derived in the next section.
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
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In the chiral representation for
, the solution space is parametrised by a
vector
, with Dirac spinor solution
where
are
Pauli 4-vectors
and
is the Hermitian matrix square-root.
Spin orientation
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Two-spinors
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]
In the Dirac representation, the most convenient definitions for the two-spinors are:
and
since these form an
orthonormal basis
with respect to a (complex) inner product.
Pauli matrices
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]
The
Pauli matrices
are
Using these, one obtains what is sometimes called the
Pauli vector
:
Orthogonality
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]
The 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
and
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
and
The normalization chosen here is such that the scalar invariant
really is invariant in all Lorentz frames. Specifically, this means
Completeness
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]
The 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
and
This follows because
which in turn follows from the anti-commutation relations for the
gamma matrices
:
with
the
metric tensor
in flat space (in curved space, the gamma matrices can be viewed as being a kind of
vielbein
, although this is beyond the scope of the current article). It is perhaps useful to note that the Dirac equation, written in the rest frame, takes the form
and
so that the rest-frame spinors can correctly be interpreted as solutions to the Dirac equation. There are four equations here, not eight. Although 4-spinors are written as four complex numbers, thus suggesting 8 real variables, only four of them have dynamical independence; the other four have no significance and can always be parameterized away. That is, one could take each of the four vectors
and multiply each by a distinct global phase
This phase changes nothing; it can be interpreted as a kind of global gauge freedom. This is not to say that "phases don't matter", as of course they do; the Dirac equation must be written in complex form, and the phases couple to electromagnetism. Phases even have a physical significance, as the
Aharonov?Bohm effect
implies: the Dirac field, coupled to electromagnetism, is a
U(1)
fiber bundle
(the
circle bundle
), and the Aharonov?Bohm effect demonstrates the
holonomy
of that bundle. All this has no direct impact on the counting of the number of distinct components of the Dirac field. In any setting, there are only four real, distinct components.
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
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It 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:
are orthogonal
and are
idempotent
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
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Charge conjugation
transforms the positive-energy spinor into the negative-energy spinor. Charge conjugation is a mapping (an
involution
)
having the explicit form
where
denotes the transpose,
is a 4×4 matrix, and
is an arbitrary phase factor,
The article on
charge conjugation
derives the above form, and demonstrates why the word "charge" is the appropriate word to use: it can be interpreted as the
electrical charge
. In the Dirac representation for the
gamma matrices
, the matrix
can be written as
Thus, a positive-energy solution (dropping the spin superscript to avoid notational overload)
is carried to its charge conjugate
Note the stray complex conjugates. These can be consolidated with the identity
to obtain
with the 2-spinor being
As this has precisely the form of the negative energy solution, it becomes clear that charge conjugation exchanges the particle and anti-particle solutions. Note that not only is the energy reversed, but the momentum is reversed as well. Spin-up is transmuted to spin-down. It can be shown that the parity is also flipped. Charge conjugation is very much a pairing of Dirac spinor to its "exact opposite".
See also
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References
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]
- ^
Jost, Jurgen (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.