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On the Physical Interpretation of the Dirac Wavefunction II On the Physical Interpretation of the Dirac Wavefunction II: The Massive Dirac Field Anastasios Y. Papaioannou [email protected] June 15, 2018 Abstract Using the language of the Geometric Algebra, we recast the massive Dirac bispinor as a set of Lorentz scalar, vector, bivector, pseudovector, and pseudoscalar fields that obey a generalized form of Maxwell’s equa- tions of electromagnetism. This field-based formulation requires careful distinction between geometric and non-geometric implementations of the imaginary unit scalar in the Dirac algebra. This distinction, which is obscured in conventional treatments, allows us to find alternative con- structions of the field bilinears and a more natural interpretation of the discrete C, P , and T transformations. 1 The massless Dirac equation We first review the use of the projection bispinor [1] to derive the multivector- valued version of the massless Dirac equation ∇ψ =0, where µ ∇≡ ∂µγ . The Dirac bispinor can be factorized into two terms, ψ = ψM w, (1) arXiv:1806.05545v1 [quant-ph] 12 Jun 2018 where the dynamical degrees of freedom are contained within the multivector field 1 ψ = f 1+ v γµ + F γµ ∧ γν + p Iγµ + gI, M µ 2! µν µ w is a fixed bispinor, and we have used notation familiar from the Geometric Algebra: 1 γµ ∧ γν ≡ (γµγν − γν γµ) 2 0 1 2 3 I ≡ γ0γ1γ2γ3 = −γ γ γ γ . 1 The multivector field ψM contains sixteen real degrees of freedom: the scalar f, four vector components vµ, six bivector components Fµν , four pseudovector components pµ, and one pseudoscalar g. These field components are all real- valued; that is, ψM is an element of the spacetime algebra, namely, the Clifford algebra Cl1,3(R). The sole purpose of the fixed bispinor w is to project these dynamical degrees of freedom onto an abstract vector space. We choose w to obey the relations γ0w = w γ2γ1w = iw, which in the Dirac basis gives it the form 1 0 w = . (2) 0 0 The massless Dirac equation, with eight real degrees of freedom, ∇ψM w =0 can be expanded into a multivector equation with sixteen real degrees of free- dom: ∇ψM =0. (3) By the nature of w, which projects γ0 and 1 onto the same spinor compo- nent, this derivation of the multivector equation from the spinor equation is not unique. We justify it, however, as being Lorentz invariant, with no mixing of, e.g., scalar and vector terms. In component form, eq. 3 becomes α βα ∂ f + ∂βF =0 α βα ∂ g + ∂βF =0 α ∂αv =0 α ∂αp =0 αβ αβ Fv + Fp =0, (4) where 1 F αβ ≡ ǫαβγδF 2! γδ αβ α β β α Fv ≡ ∂ v − ∂ v 1 F αβ ≡ ǫαβγδ(∂ p − ∂ p ). p 2! γ δ δ γ There is no mixing between even-grade (scalar, bivector, and pseudoscalar) and odd-grade (vector and pseudovector) fields. For simplicity, we will assume that the odd fields vanish, leaving eight degrees of freedom: 1 ψ = f 1+ F γµ ∧ γν + gI, (5) M 2! µν 2 or, in spinorial form, f − iF12 −iF23 + F31 ψ = . (6) ig − F30 −F10 − iF20 1.1 Lorentz transformations In the spacetime algebra, we interpret the gamma matrices {γµ} as matrix rep- resentations of physical basis vectors. Under a Lorentz transformation between two reference frames, these basis vectors transform as γµ → γ′µ = Rγµ R, where e 1 R = exp(− ω γµ ∧ γν ) 4 µν is the half-angle / half-rapidity Lorentz transformation operator, and R is the multivector reverse of R, in which all constituent vectors in R are reverse- e ordered, i.e., (γµ ∧ γν)∼ = γν ∧ γµ = −(γµ ∧ γν ). The reversed operator 1 R = exp(+ ω γµ ∧ γν ) 4 µν is thus the multiplicative inversee of R, with ωµν → −ωµν . Higher-grade ele- ments inherit their Lorentz transformation properties from the vectors compris- ing them: γ′µ1 γ′µ2 ··· γ′µn = (Rγµ1 R)(Rγµ2 R) ··· (Rγµn R) = R(γµ1 γeµ2 ··· γµen )R. e That is, elements of all grades transform in the samee manner: M → RMR. To derive the corresponding expressione for the transformation of a multi- vector component under a change of reference frame, we contract ψM with the appropriate basis element in the new reference frame, e.g., ′µ ′µ µ v = hψM γ i = hψM (Rγ R)i, where hMi denotes the scalar term of the multivectore M. By the cyclic property of products in a scalar term, µ µ hψM (Rγ R)i = h(RψM R)γ i, e e 3 which is the contraction of the inverse-transformed multivector onto γµ. Similar rules hold for the other basis elements. Under a change of reference frame, then, we can write ψM → SψM S, where the transformation operator is e 1 S = R = exp(+ ω γµ ∧ γν), 4 µν e so that each term in SψM S is the Lorentz-transformed component: e 1 Sψ S = f ′1+ v′ γµ + F ′ γµ ∧ γν + p′ Iγµ + g′I. M µ 2! µν µ e Given the one-sided transformation rule for the Dirac bispinor: ψ → ψ′ = Sψ, we find the corresponding one-sided transformation rule also holds for the pro- jection bispinor w: ψM w → SψM w = (SψM S)(Sw). e 1.2 Bilinears The factorization of the Dirac bispinor allows us to construct bilinears solely in terms of multivectors. For a general multivector element M, the bilinear ψMψ takes the form † † 0 ψMψ = (w ψM γ )M(ψM w) † 0 = w γ ψM MψM w 0 = (γ ψMeMψM )11, where we have written the Hermitian conjugatee in a basis-independent form using the reversal operation: † 0 0 ψM = γ ψM γ . In the Dirac basis, only 1, γ0, γ2γ1, and γ2eγ1γ0 have non-zero (1, 1) components. The bilinear expression therefore projects out the terms proportional to these four basis elements: 0 0 2 1 2 1 0 ψMψ = h(γ ψM MψM )(1 + γ − iγ γ − iγ γ γ )i. (7) For example, the vector bilineare jµ for the massless Dirac field is µ 0 µ 0 2 1 2 1 0 ψγ ψ = h(γ ψM γ ψM )(1 + γ − iγ γ − iγ γ γ )i 0 2 1 2 1 0 0 µ = hψM (1e + γ − iγ γ − iγ γ γ )γ ψM γ i 0 µ = h(ψM γ ψM )γ i, e e 4 µ 0 which gives the γ component of the vector quantity j = ψM γ ψM . In compo- nent form, we have e 2 2 2 2 0 i j = (f + g + E~ + B~ )γ − 2(fE~ + gB~ + E~ × B~ )iγ . (8) To draw out the similarity to electromagnetism, we have written the bivector components as i0 Ei = F 1 B = − ǫijkF jk. i 2 Via similar methods, the pseudovector spin angular momentum density is i S0ij = ψγ0(γi ∧ γj )ψ 2 i = w†γ0ψ γ0(γi ∧ γj )ψ w 2 M M 1 0 e 0 0 2 1 2 1 0 = hγ ψ γ (γi ∧ γj )ψ i(1 + γ − iγ γ − iγ γ γ )i 2 M M 1 0 e 0 2 1 2 1 0 0 = hγ ψ γ (γi ∧ γj )ψ (γ γ + γ γ γ + i1+ iγ )i 2 M M 1 e 2 1 0 0 = h(ψ γ γ γ ψ )γ (γi ∧ γj )i. 2 M M e This gives the (0ij) component of the pseudovector quantity 1 S = ψ γ2γ1γ0ψ . (9) 2 M M e For i = 1, j = 2, we have the “z-component” of the spin, which in component form is: 012 1 2 2 2 2 2 2 S = (f + g − E~ − B~ +2E3 +2B3 ). 2 Note that the factor of i in the bilinear effectively serves as multiplication by γ2γ1: 0 2 1 2 1 0 2 1 0 2 1 2 1 0 i(1 + γ − iγ γ − iγ γ γ )= γ γ (1 + γ − iγ γ − iγ γ γ ), allowing us to implement factors of the imaginary unit scalar using an element of the real algebra Cl1,3(R): 2 1 iψ = iψM w = ψM γ γ w. 1.3 The field and operator interpretations We emphasize the distinction between our “field interpretation” of the multi- vector ψM and the conventional “operator interpretation” in other treatments of the Dirac field in the spacetime algebra [2], [3], [4]. In the field interpretation, 5 ψM describes dynamic field degrees of freedom. The massless Dirac equation, for example, describes the dynamic behavior of one scalar, one pseudoscalar, and six bivector flux fields: 1 ψ = f 1+ F γµ ∧ γν + gI, M 2! µν which obey a generalized form of Maxwell’s equations. The projection bispinor is merely a mathematical device that allows for simplified calculations within matrix representations of the underlying algebra, by projecting those degrees of freedom onto an abstract vector space. In other formulations of the Dirac equation in the spacetime algebra, the multivector is instead treated as a dynamic transformation operator that acts upon a constant reference bispinor. In this operator interpretation, ψM trans- forms the bispinor and other constant reference frame basis elements. Bilinears are interpreted as transformed basis elements, not as derived quantities that are quadratic in a more fundamental field quantity. In the field interpretation, the field is fundamental and the bilinears are derived quantities. 2 The massive Dirac equation The Klein-Gordon equation for a real scalar field f is 2 f = −ω0 f, 2 2 2 where ≡ ∂tt − c ∇~ and ω0 = mc /~. To derive the corresponding first- order equations, we first extend f to an even-grade multivector, by introducing bivector and pseudoscalar fields: 1 f → ψ = f 1+ F γµ ∧ γν + gI, e 2! µν each component of which also obeys the Klein-Gordon equation: 2 ψe = −ω0 ψe.
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