Computational Determination of the Dirac-Theory Adjunctator

Abstract

A number of particle properties stem from the use of 𝛾⁰ as adjunctator (Bargmann-Pauli) in the Dirac theory (spin alignment, Dirac current, etc.). The early motivations for accepting 𝛾⁰ as adjunctator were representation-dependent, mildly bearing relation to the actual conditions forcing 𝛾⁰ as adjunctator. Representation-independent approaches to the physical predictions of the Dirac equation are somewhat new, here presented as being the reasons for 𝛾⁰ as adjunctator of the Dirac theory, together with the essential role of the latter in the physical aspects of the theory.

1. Introduction

The definition of the Dirac operators (𝛽,𝛼) as 𝛾⁰(1,𝛾), satisfying [𝛾^𝜇,𝛾^𝜈]₊=2𝑔^𝜇𝜈⋅𝟏, that is, a Clifford algebra 𝒞𝑙_1,3(𝐶), does not include any information on how adjoint operators are constructed. In representation-dependent form, as matrices, the adjoints are constructed as (𝛾^𝜇†)_𝛼,𝛽=(𝛾^𝜇)^∗_𝛼,𝛽; however, such representation-dependent solutions also include representation-specific artifacts, often nonphysical. The importance of presenting properties in representation-independent form, stemming from minimal information in the abstract definition of the foundations, thus cannot be emphasized enough.

The adjunctator (algebra element that constructs the adjoint) is present in a number of particle properties (current definition, spin alignment, etc.). In Dirac theory, this is the Bargmann-Pauli [1–3] adjunctator 𝛾⁰ and its unicity is crucial to knowing that the physical quantities used are well defined.

The initial demand in determining how the adjoint is constructed for (𝛾⁰, 𝛾) was from the Dirac-1928 equation itself as follows: 𝐄=𝐇=𝛼⋅𝐩+𝛽𝑚𝑐_|𝜓, (1) where 𝐄 is the energy, 𝐇 the Hamiltonian, (𝛽,𝛼) Dirac’s initial matrices, and 𝜓 the system’s wave function. Since 𝐄=𝐄^† it seemed natural to ask that 𝐇=𝐇^†. This is rather ad-hoc, all that the above relation has to offer being Im{⟨𝐇^†−𝐇⟩_𝜓}=0.

Leaving such “physics” arguments behind, the real aspects can be now explored. Little attention was given to this topic; however Pauli [3] did have interest in it at the time. His arguments though representation dependent (with one minor inaccuracy) offer the only explanation to date on this topic. The only representation-independent study [4, 5] is about the unicity of the adjunctator together with a pleiad of related studies [6, 7].

In the Dirac algebra (of the 𝛾^𝜇 matrices), the adjoint operators form another representation of the same algebra, as the arguments of Pauli [3] show the following: 𝛾^𝜇†,𝛾^𝜈†₋=−𝛾^𝜇,𝛾^𝜈₋^†=2𝑔^𝜇𝜈⋅𝟏⟹𝛾^𝜇†=𝐁𝛾^𝜇𝐁⁻¹. (2) The two representations are thus related through a similarity transformation 𝐁 termed adjunctator or hermitiser [1, 2], with the following properties:(i)(𝛾^𝜇†)^†=𝛾^𝜇=𝐁^†−1𝛾^𝜇𝐁^†=(𝐁^†−1𝐁)𝛾^𝜇(𝐁^†−1𝐁)⁻¹⇒[𝐁^†−1𝐁,𝛾^𝜇]₋=0 for all 𝜇=0,3, respectively, 𝐁^†=𝜆𝐁. (ii)(𝐁^†)^†=(𝜆𝐁)^†=|𝜆|²𝐁=𝐁⇒|𝜆|=1. The reasons for the consecrated 𝐁=𝛾⁰ must thus lie in the mathematical foundations of the bispinor space and cannot be collected from the Lagrangian conditions already using 𝛾⁰ as adjunctator, other ad hoc conditions (shown in the beginning), or representation-specific arguments leading to generalisations not pertinent to the problem.

The representation-independent conditions applicable are(i)self-adjointness: any 𝐗²=𝟏 has real eigenvalues; hence 𝐗=𝐗^†. For example, 𝐗=𝜎_𝑖,𝛾⁰,𝛾⁵, and so forth; (ii)positive definiteness: for any 𝐗, its norm squared ‖𝐗𝜓‖²=⟨𝐗𝜓∣𝐗𝜓⟩≥0; hence 𝐁 must be such that 𝐗^†𝐗≥0, ∀𝐗. Respectively they are the mathematical translation of understanding the scalar-product. The early arguments of Pauli [3] relied on counting representation-dependent anti-/symmetric matrices. This eludes the above two conditions and leads to one minor misconclusion on 𝐁.

2. Structure of the Scalar Product

Let Γ=𝛾^𝜇𝛾^𝜈⋯𝛾^𝜌; then Γ^†=𝐁𝛾^𝜌⋯𝛾^𝜈𝛾^𝜇𝐁⁻¹=sign⋅𝐁Γ𝐁⁻¹. It is useful thus to define Γ=^def𝐁⁻¹Γ^†𝐁, (3) with the trivial action of complex conjugation on scalars and a generalisation of this on operators as follows: (i)𝐗𝐘=𝐘𝐗, (ii)𝐗⁻¹=𝐗⁻¹, (iii)𝐑+𝑖𝐂=𝐑−𝑖𝐂, for 𝐑 and 𝐂 “standard”, that is, 𝐑=𝐑 and 𝐂=𝐂; (iv)ΓΓ=ΓΓ; that is, ΓΓ is standard; (v)𝜆𝐗+𝜇𝐘=𝜆𝐗+𝜇𝐘; (vi)if 𝐑+𝑖𝐂=0; then both 𝐑 and 𝐂 are zero since 0=𝐑−𝑖𝐂=0. With this notation, it is easier to understand the action of 𝐁 as follows: Γ^†=𝐁Γ𝐁⁻¹ (4) and to further discover the adjunctator: 𝐁^†=𝜆𝐁=𝐁𝐁𝐁⁻¹⇒𝐁=𝜆𝐁. The operator 𝐁 is comprised of a standard and antistandard part: 𝐁=𝐑+𝑖𝐂, where both 𝐑 and 𝐂 are standard. Therefore 𝐁=𝜆𝐁 translates to 𝐑−𝑖𝐂=𝜆𝐑+𝑖𝜆𝐂, which takes into account that 𝜆=𝜆_𝑅+𝑖𝜆_𝐶 and also in view of property (vi) becomes. 𝜆_𝑅−1−𝜆_𝐶𝜆_𝐶𝜆_𝑅𝐑+1𝐂=0. (5) Respectively, the determinant must vanish (which for |𝜆|=1, the case at hand, actually does). The condition from (5) is that 𝐑∼𝐂, or 𝐁=𝑧𝐑 where 𝑧 is a unit complex number relevant only for aesthetical purposes as follows: Γ^†=(𝑧𝐁)Γ(𝑧𝐁)⁻¹=𝐁Γ𝐁⁻¹. (6) The only effect of 𝑧 is that of choice: with 𝑧=1, 𝐁^†=𝐁, with 𝑧=𝑖, 𝐁^†=−𝐁, with 𝑧=𝑒^𝑖𝜃/2, 𝐁^†=𝑒^𝑖𝜃𝐁—therefore 𝑧 can be chosen 1, hence 𝐁 standard and 𝐁^†=𝐁. The property of significance is that the adjunctator is, up to a unit complex multiplier, standard. The arguments in [3] of counting symmetric and antisymmetric matrices, though reaching the same conclusion, forbid 𝐁^†=−𝐁 which, as shown above, could have served equally well.

3. Self-Adjoint Operators

The simplest proof from this point on is by looking at the generators of the Dirac algebra-(𝜎,𝑔), where 𝜎=𝛾⁰𝛾𝛾⁵ and 𝑔=(𝛾⁵,−𝑖𝛾⁰𝛾⁵,𝛾⁰); the two sets defining two isomorphic halves of the space [8] are shown as follows: 𝜎×𝜎=2𝑖𝜎,with𝜎²_𝑖=𝟏,𝑔×𝑔=2𝑖𝑔,with𝑔²_𝑖=𝟏. (7) The two halves of the space admit various other representations, such as (𝟏−𝛾⁵)𝜎/2 and (𝟏+𝛾⁵)𝜎/2; however, the one presented is the most convenient for the task at hand. The endomorphism between the two halves is 𝐒=𝟏+𝜎𝑔=𝟏+𝛾⁰𝛾¹−𝑖𝛾²+𝑖𝛾⁰𝛾¹𝛾².

The two halves seem identical; however from the adjunctation point of view they differ as follows: 𝜎₁,𝜎₂,𝜎₃=𝜎₁,𝜎₂,𝜎₃,𝑔₁,𝑔₂,𝑔₃=−𝑔₁,−𝑔₂,𝑔₃. (8) Encouraged by the fact that 𝜎²_𝑖=𝟏 and 𝑔²_𝑖=𝟏, we explore under what conditions 𝜎^†=𝜎 and 𝑔^†=𝑔.

For the first half of the space, we seek an adjunctator in the form of 𝐵=𝜆+𝑎𝜎, where [𝜆,𝜎]₋=0 and [𝑎,𝜎]₋=0. With some loss of generality, we let [𝜆,𝑎]₋=0 and 𝑎×𝑎=0, for the sake of exemplifying the mechanisms in place. Under this adjunctator; 𝜎^†=2𝜎+𝜆²−𝑎²𝑎²𝜎_⟂−𝜆𝑎×𝜎. (9) For 𝑋=𝑏𝜎, in order that 𝑋^†𝑋≥0, it is evident that 𝑎=0, leaving 𝜆=scalar+LIN{𝑔}.

For the second half of the space, similar arguments follow, only that in this case the 𝑔_𝑖’s are not similar to the 𝜎_𝑖’s. Further restricting generality, assume 𝑎=𝑎𝑒₃; then 𝑔^†=2𝑔−𝜆²−𝑎²𝜆²𝑔_⟂−𝜆𝑎×𝑔 (10) The role of 𝑎=0 in the first half transfers here to 𝜆=0.

It can be seen that 𝐵=𝑔₃ performs 𝜎^†=𝜎 and 𝑔^†=𝑔, which for any operator in the space satisfies 𝑋^†𝑋≥0.

Conversely, without proof, if 𝜎^†=𝜎 and 𝑔^†=𝑔, then any operator in the space satisfies 𝑋^†𝑋≥0.

Let 𝑋=𝛾⁰𝑋𝛾⁰ be an operation with the same properties as 𝐗, mainly just complex conjugating the coefficients in a sum of operators. If 𝐀=𝐁𝛾⁰, then 𝑋^†=𝐀𝑋𝐀⁻¹ and 𝜎^†_𝑖=𝐀𝜎_𝑖𝐀⁻¹=𝐀𝜎_𝑖𝐀⁻¹=𝜎_𝑖,𝑔^†_𝑖=𝐀𝑔_𝑖𝐀⁻¹=𝐀𝑔_𝑖𝐀⁻¹=𝑔_𝑖, (11) as assumed, hinted by the fact that 𝜎_𝑖 and 𝑔_𝑖 squared are unity and have real eigenvalues. This implies that 𝐀,𝜎_𝑖−=0,∀𝑖=1,3,𝐀,𝑔_𝑖−=0,∀𝑖=1,3, (12) and in turn 𝐀𝛾⁰𝛾^𝑖𝛾⁵=𝛾⁰𝛾^𝑖𝛾⁵𝐴,𝐀𝛾⁰=𝛾⁰𝐀,𝐀𝛾⁵=𝛾⁵𝐀. (13) Respectively, both [𝐀,𝛾^𝑖]₋=0 and [𝐀,𝛾⁰]₋=0; that is, [𝐀,𝛾^𝜇]₋=0, ∀𝜇=0,1, or 𝐀=𝟏.

All these exemplifications point to the asymmetry—with respect to adjunctation—of the two halves of the space and to the fact that 𝑔₃=𝛾⁰ may be the Bargmann-Pauli adjunctator.

4. Positive Definiteness of 𝑋^†𝑋

A more involved argument is that 𝐁 should be such that 𝑋^†𝑋 is positive definite for any 𝑋; that is: 𝐀𝑋𝐀⁻¹𝑋≥0, which is the matrix version of understanding the scalar product. Since (𝜎,𝑔) have the same properties, and the arbitrariness of 𝑋 imposes equal constraints on both, that is, any property holding for 𝑋(𝜎,𝑔) must also hold for 𝑋(𝑔,𝜎), it follows that 𝐀 is symmetric in (𝜎,𝑔) as follows: 𝐀=𝑏⋅𝟏+𝑎𝜎+𝑔+𝑀_𝑖𝑗𝜎_𝑖𝑔_𝑗+𝜎_𝑗𝑔_𝑖. (14) In the above, 𝑎 and 𝑀_𝑖𝑗 are unjustifiable anisotropies; hence 𝑎=0 and 𝑀_𝑖𝑗=𝛽𝛿_𝑖,𝑗; therefore 𝐀 reduces to 𝐀=𝛼⋅𝟏+𝛽𝐒, with 𝐀^−𝟏=(𝛼−𝛽𝐒)/(𝛼²−4𝛽²), where 𝐒^𝟐=4⋅𝟏. The 𝐀𝑋𝐀⁻¹𝑋≥0 inequality becomes 𝐀𝑋𝐀⁻¹𝑋=(𝛼²−4𝛽²)⁻¹×𝛼²𝑋_𝜎−𝛽²𝑋_𝑔+𝛼𝛽𝐒𝑋_𝜎−𝑋_𝑔𝑋_𝜎≥0, (15) where 𝑋_𝜎=𝑋(𝜎,𝑔) and 𝑋_𝑔=𝑋(𝑔,𝜎), 𝐒 playing the go-between role: 𝐒𝑋_𝜎=𝑋_𝑔𝐒 and 𝑋_𝜎𝐒=𝐒𝑋_𝑔. Let 𝑋_𝜎=𝑋_𝑔=𝜆+𝜇𝐒; then (15) (using 𝐒=𝐒) becomes 𝐀𝑋𝐀⁻¹𝑋=𝛼²−4𝛽²⁻¹×𝜆𝛼²−𝛽²+𝜇𝛼²+𝛽²+8×𝜇𝛼𝛽𝜆+𝜇𝐒≥0. (16) Expanding the above, 𝐀𝑋𝐀⁻¹𝑋=𝛼²−4𝛽²⁻¹×𝜆𝜇+𝜆𝜇𝛼²𝐒+𝜆𝜇−𝜆𝜇𝛽²𝐒+|𝜆|²𝛼²−𝛽²+4𝜆𝜇+𝜇𝜆𝜇𝛼𝛽+8𝛼𝛽²𝐒+4𝜇²𝛼²+𝛽²+4𝜆𝜇−𝜆𝜇𝛼𝛽≥0, (17) where the imaginary boxed terms disappear for 𝛽=0; hence 𝐀∼𝟏 and B=𝛾⁰ (qed). Note that the expectation values of 𝐒=𝟏+𝜎⋅𝑔 can be 1/4 and 7/4, stemming from the product of two independent 1/2 angular momenta.

In general form (without the anisotropy argument), the proof ends annihilating all 𝑎 and nondiagonal 𝑀_𝑖𝑗 coefficients for the same reason above: the existence of imaginary terms in a relation that needs to be real and positive.

The unicity issue has been addressed by direct construction, although proof in this respect can be found in [4, 5].

Another aspect of interest is the possibility of using the bar in place of (adjunctation)† on the helicity-positive subspace as follows: 𝜓∣𝜙=^def𝐂𝜓∣𝐂𝜙. (18) In this case, 𝐁=𝐂𝐂>0, the 𝐂 operator taking the in-between bearing role of 𝛾⁰. Evidently, any operator satisfying 𝐒𝐒=𝟏 invaries the new scalar product of (𝜓∣𝜙). The particularity of this context is that any 𝐂 satisfies the scalar product and norm conditions (i.e., absolutely arbitrary scalar product), leading from the physics point of view to arbitrary many Lorentz invariant conserved currents.

5. Dirac Currents

Another interesting aspect is that two Lorentz invariant quantities which include charge density ⟨𝜌⟩=|𝜓(𝑥)|² can be defined: 𝜌𝐁𝛾^𝜇 and 𝜌𝐁𝛾^𝜇𝛾⁵. Of these two, only the former is conserved: 𝜕_𝜇⟨𝜌𝐁𝛾^𝜇⟩_𝜓=0.

It is clear that if the adjunctator 𝐁 is not unique, more than one such conserved Lorentz invariant quantity can be defined. The transformation properties under space-, time-, and space-time inversion for 𝑗^𝜇=⟨𝜌𝛾⁰𝛾^𝜇⟩_𝜓 are 𝑗^𝜇𝜓^𝐏𝜇𝜓,𝑗⟶−𝑗_𝜇𝜓^𝐓𝜇𝜓,𝑗⟶−𝑗_𝜇𝜓⟶^𝐏T𝑗^𝜇𝜓, (19) where 𝐔^†_𝑃=−𝐔_𝑃, 𝐔^†_𝑇=−𝐔_𝑇 and 𝐔^†_𝑃𝑇=𝐔_𝑃𝑇 were evidently included. The notation 𝑎𝜇 refers to 𝑔^𝜇𝜈𝑎_𝜈. Above, 𝑗^𝜇 is not an operator, but rather the product: 𝑗^𝜇=⟨𝜌⟩_𝜓⋅⟨𝐯^𝜇⟩_𝜓. Its transformation properties seem to differ from textbook electrodynamics by the minus sign. This comes from the fact that charge density is better defined as a pseudoscalar under both space- and time-inversion [9–12]. Incidentally, this also solves the problem of not finding a representation-independent charge-conjugation operator 𝐂, since time reversal does this automatically for all currents, without the need of a specific charge-conjugation operator for each current in its own space.

6. Conclusion

It has been shown in representation-independent form that the adjunctator for the Dirac space is the 𝛾⁰ matrix. This is an important result, as the adjunctator enters the definition of the Dirac current and multiple adjunctators would imply the possibility of defining multiple currents, which, incidentally, is possible for the positive helicity subspace.

As the adjunctator plays also the role of parity transformation, its unicity is essential to the theory. Furthermore, together with time inversion, it defines complete coordinate inversion, which reverses all currents—without the need for a charge-conjugation operator. The latter is impossible to define in representation-independent form, which in itself is a statement over why it does not exist and why it is important to have representation-independent results.

Acknowledgments

This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0323.