On the physical interpretation of the Dirac wavefunction
Using the language of the Geometric Algebra, we recast the massless Dirac bispinor as a set of Lorentz scalar, bivector, and pseudoscalar fields that obey a generalized form of Maxwell’s equations of electromagnetism. The spinor’s unusual rotation symmetry is seen to be a mathematical artifact of the projection of these fields onto an abstract vector space, and not a physical property of the dynamical fields themselves. We also find a deeper understanding of the spin angular momentum and other Dirac field bilinears in terms of these fields and their corresponding analogues in classical electromagnetism.
The interpretation of the wavefunction remains a mystery at the heart of quantum theory. While the practical success of quantum mechanics and quantum field theory is undeniable, the Dirac bispinor, and the matrix-valued algebra which acts upon it, lacks a clear physical interpretation. In the present work, we use the Geometric Algebra to point the way towards a clearer understanding of the physical structure and transformation properties of the Dirac bispinor.
We begin in section 2 by summarizing the Geometric Algebra as it applies to -dimensional relativistic physics. Section 3 then applies these concepts to the massless Dirac equation, and develops a formulation of the Dirac bispinor in which the dynamical and spinorial degrees of freedom are distinct. With this separation of the degrees of freedom, the Dirac equation takes the form of Maxwell’s equations of classical electromagnetism, with electric and magnetic current densities. Section 4 then constructs the Dirac field bilinears, with special emphasis on their parallels with the bilinears of classical electromagnetism. Finally, in section 5 we analyze the Lorentz transformation properties of the fields, showing how the dynamic and spinorial degrees of freedom transform separately.
2 Summary of the Geometric Algebra
In this section, we briefly summarize the Geometric Algebra and its application to relativistic physics. For a much deeper mathematical development, and a broader range of applications in classical and modern physics, we refer the reader to works by Hestenes ,  and Doran and Lasenby .
The algebra is constructed from four basis vector elements , subject to the anticommutation law
where is the scalar identity element of the algebra. We use the metric with signature , so that . The anticommutation law defines the symmetric inner product of two vectors and directly implies that any two orthogonal vectors anticommute. The antisymmetric wedge or outer product of two vectors is:
Aside from the new terminology and notation for the inner and outer product, these operations are familiar from the Dirac algebra and its associated matrices . We emphasize, however, that although the algebraic structure is the same, we are not working within a (matrix) representation, but with algebraic elements interpreted as unit basis vectors. The outer product is a new two-dimensional, or grade-2, element of the algebra, the bivector. (The scalar and vector elements are grade-0 and grade-1, respectively.) The full associative, non-commutative product of two vectors is the sum of their inner and outer products:
The product of two vectors and will in general contain both scalar and bivector terms:
The inner product is the familiar scalar product of two vectors:
while the outer product is the relativistic analogue of the vector cross product :
Unlike, however, the vector cross product , which multiplies two vectors to form a third (axial) vector, the bivector is a new two-dimensional element of the algebra, which is better viewed not as a vector but as an oriented area in the plane defined by and .
Six linearly independent bivectors can be constructed from the basis vectors:
(We have explicitly written these bivectors using the wedge operator to indicate that the antisymmetric outer product has been used. When two vectors and are orthogonal, the outer product is equivalent to the full product . In cases where this orthogonality is clear, e.g., for a timelike bivector , we will often simplify our notation and write the product as .)
The bivectors inherit their multiplicative properties from the vectors. For example, the square of the bivector is
Similar calculations for the other bivector basis elements show that the “timelike” bivectors of the form all square to , while the “spacelike” bivectors square to .
Likewise, the multiplication of vectors with bivectors will also follow from the multiplicative properties of the constituent vectors. The product of with , for example, is the vector :
while the product of with the same bivector is the vector :
Multiplication by has thus rotated the elements clockwise by in the plane:
We will make further use of this property of bivectors when analyzing Lorentz transformations.
The product of with , however, is not a vector, but a new grade-3 element, a pseudovector:
Here we have extended the outer product to higher grades: for vectors , , and , is the grade-3 term(s) of the product . More generally, the outer product of vectors, will be the grade- term(s) of the product . By the anticommutation properties of its constituent vectors, the outer product is fully antisymmetric upon interchange of any pair of vector indices. This full antisymmetry also guarantees that the outer product is associative, e.g., .
There are four linearly independent pseudovectors:
The three pseudovectors containing all square to , e.g.,
while the remaining pseudovector squares to :
There is, finally, one unique grade-4 pseudoscalar,
which squares to :
In total there are 16 linearly-independent basis elements of the algebra: 1 scalar, 4 vectors, 6 bivectors, 4 pseudovectors, and 1 pseudoscalar. Any vector multiplying the pseudoscalar will contract with one of its constituent basis vectors, so that no elements of grade 5 or higher are possible. The basis contains six square roots of :
and ten roots of :
2.1 Grade Parity, Duality, and the Multivector
We call even those quantities that have an even grade, namely the scalar, bivectors, and pseudoscalars; those with odd grade (the vectors and pseudovectors) are called odd. This concept of “grade parity” will be extremely useful when analyzing bilinear quantities: any product containing only even quantities (or an even number of odd quantities) will necessarily also be even, and any product containing an odd number of odd quantities will necessarily be odd. This allows us to analyze complicated products and to determine immediately, based on the grade parity, which terms exist and which are necessarily zero.
In later sections we will make extensive use of a few important properties of . As previously noted, it is a root of . Also, because it contains one factor of each basis vector, it anticommutes all odd elements and commutes with all even elements (including itself). Finally, serves as a duality operator, both between the spacelike and timelike bivectors:
and between the vectors and pseudovectors:
We will use this duality operation to greatly simplify our calculations below.
A general element of the algebra, the multivector, is a sum of scalar, bivector, pseudovector, and pseudoscalar terms:
By analogy with the bivector field of electromagnetism, we can write and for the bivector field components, and simplify the pseudovector and pseudoscalar field components:
With these changes, the multivector takes the form
In the Geometric Algebra, the coefficients , , , , , and take only real values. As we will see below, any imaginary values will be interpreted in terms of the basis elements that are roots of .
2.2 Grade Extraction, the Hermitian Adjoint, and Reversal
For a general multivector , we will often wish to isolate terms of a particular grade. denotes the term(s) with grade . If the subscript is omitted, the scalar (grade-0) term is implied: . One property of the scalar term that will prove extremely useful is that a product of multivectors is unaffected by cyclic permutations:
As mentioned previously, we can generalize the inner and outer products between vectors to higher-grade objects. For multivectors and with respective non-zero grades and , the product will contain terms with grades , , etc., up to . The inner product denotes the term(s) with minimum grade :
and the outer product will be defined as the term(s) in the product with the maximum grade :
The Hermitian adjoint of a product of vectors is the reverse-ordered product of the Hermitian adjoint of the individual vectors:
In both the Weyl and Dirac bases, the Hermitian adjoint changes the sign of , leaving unchanged:
The corresponding operation in the Geometric Algebra (, ) is equivalent to a spatial reflection through the origin. This operation can be written as a two-sided transformation:
The Hermitian adjoint of a product of basis vectors then becomes:
where the reversal operation reverses the order of all constituent vectors in the multivector .
The net result of Hermitian conjugation depends on two factors: parity under the reversal operation, and spatial parity. Upon reversal, the scalars, vectors, and pseudoscalars are even, whereas the bivectors and pseudovectors are odd:
A general multivector will therefore have mixed parity under reversal.
Under spatial reflection, elements composed of an even number of spacelike vectors, namely , , , and , are positive; those composed of an odd number (, , ) are negative. If an element has the same parity under both reversal and spatial reflection, then it will be its own Hermitian conjugate. The self-conjugate basis elements are:
The basis elements that have opposite parities under reversal and spatial reflection will change sign under Hermitian conjugation:
The self-conjugate elements are the roots of , and the elements that change sign under conjugation are the roots of , making Hermitian conjugation the natural extension of the complex conjugation operation to the Geometric Algebra.
3 The massless wave equation
Using the Geometric Algebra, we can write the “square root” of the second-order differential operator in terms of the first-order, vector-valued operator :
For a scalar field subject to the massless wave equation
we expand the number of fields from the single scalar to a multivector with 16 field degrees of freedom:
which is subject to the multivector-valued first-order differential equation
The vector derivative of the scalar is a vector-valued quantity:
The bivector field will have vector and pseudovector derivative terms:
The derivative of the pseudoscalar will have pseudovector terms:
The first-order differential equation can therefore be separated into vector-valued and pseudovector-valued equations:
There is no coupling of the scalar, bivector, and pseudoscalar fields to any vector or pseudovector fields that may also be present. We will therefore assume that the vector and pseudovector fields are zero.
In terms of these components, the above equations yield:
That is, the first-order wave equation is equivalent to Maxwell’s equations of electromagnetism, with playing the role of electric current and that of magnetic current. The multivector has eight components, and satisfies the first-order massless wave equation, as does the (massless) bispinor in the Dirac algebra.
3.1 Lorentz transformations
The Geometric Algebra provides a natural unified language for describing transformations. For example, the rotation of the vector by angle in the plane can be written
where the multivector operator is
This transformation could equivalently have been written in terms of an operator that right-multiplies the vector:
or as a two-sided operation, with half of the transformation performed by each operator:
where is the reverse of . The benefit of this two-sided operation is that all elements of the algebra transform in the same way, including higher-grade elements as well as vectors that are not in the plane of rotation:
where we have made use of the important fact that is the inverse of .
Lorentz boosts can be written in the same way, as a “rotation” with a timelike bivector:
and a general proper Lorentz transformation can be written as a product of a boost with rapidity and a rotation through angle :
where is the unit normal to the plane of rotation and is the direction of the boost. (The extra negative sign in the factor containing appears because we have chosen to write the spacelike bivector in the dual form .)
3.2 The Dirac bispinor
We now construct a correspondence between the multivector and bispinor solutions. Each contravariant basis vector in the Geometric Algebra corresponds is represented by a matrix (taking care to note changes of sign when changing covariant spatial vectors to contravariant ones). The bivector and higher-grade products all carry over directly:
A matrix-valued solution to the massless wave equation follows immediately:
In short, replacing each basis vector with its corresponding gamma matrix yields a matrix-valued solution to the massless wave equation, with scalar, bivector, and pseudoscalar degrees of freedom. The familiar Dirac bispinor solution, however, is not represented by a square matrix but by a column vector upon which the Dirac matrices act. We seek therefore a method of “projecting” the above matrix solution onto the vector space, by choosing a constant projection column vector or projection bispinor , which right-multiplies the matrix solution to give the bispinor solution:
The choice for the constant two-component spinors and will be informed by the chosen basis (e.g., Dirac or Weyl). Below, we will use the Dirac basis.
To find and we first note that the imaginary scalar has no obvious implementation in the Geometric Algebra, i.e., there is no linear combination
with real components , , , and , that commutes with all other elements and also squares to . The implementation of will therefore involve some basis-specific operation. To find such an operation, we note that factors of can always be shifted to multiply the projection bispinor :
We seek an operation that will multiply the projection spinors and by :
In the Dirac basis,
where , , and are the identity, zero, and Pauli matrices. Of these matrices, only , , , and are block diagonal, and of these only , , , and also have imaginary entries. We choose , so that
If and , then we have a candidate for . The exact choice of and will be determined by our additional requirement that be an eigenvector of :
In the Dirac basis,
Therefore, and , and the projection bispinor becomes
The bispinor now takes the form
4 Dirac Bilinears
With the above choice of basis and projection bispinor, bilinears can easily be rewritten in terms of the multivector field components. The general procedure for constructing Dirac bilinears is as follows. The Dirac adjoint of the Dirac bispinor is
For an operator , the bilinear now takes the form
With the chosen projection bispinor , the product singles out the component of the matrix :
In the Dirac basis, the only matrices with a component are , , , and . Taking the component of a matrix is therefore equivalent to isolating the terms proportional to these four basis elements:
(In our matrix representation, extraction of the grade-0 terms can be accomplished by a trace operation: .)
Because all bilinears of physical interest are constructed to be real quantities, the imaginary and terms will be zero, simplifying the expression to:
Furthermore, we are interested only in operators that correspond to physical quantities with well-defined transformation properties. The bilinears will therefore always be constructed so that is either purely even or purely odd, and only one of the terms in (18) will be non-zero.
4.1 The scalar and pseudoscalar bilinears
For the Lorentz scalar bilinear, we have
The operator is the product of the (odd) vector with two even multivectors. It is therefore odd, and the scalar is necessarily zero, leaving only:
By the cyclic property of the scalar term,
In terms of the field components of , the scalar becomes
Setting , we recover the familiar Lorentz scalar of classical electromagnetism:
The pseudoscalar bilinear is
Here is again odd, so that , and
which isolates the pseudoscalar term of . In terms of the field components,
When , we recover the familiar electromagnetic pseudoscalar:
4.2 The vector and pseudovector bilinears
Applying the same procedure to the vector current , our bilinear is:
With an even product, is identically zero, leaving
In other words, this quantity projects the component of the vector . We can see that is a vector, because it is odd and positive under reversal:
Multiplying out the product in terms of the field components,