# Full text of "The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization"

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N92-22088 THE HAMILTONIAN STRUCTURE OF DIRAC'S EQUATION IN TENSOR FORM AND ITS FERMI QUANTIZATION Frank Reifler and Randall Morris Government Electronic Systems Division General Electric Company Moorestown, NJ 08057 ABSTRACT Currently there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quan- tization. In this paper, we demonstrate that the bis- pinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Pois- son brackets as commutators for the Heisenberg op- erators directly leads to Fermi quantization without the use of bispinors. 1. CLASSICAL DERIVATION We apply the double covering map from bispinors to their tensor equivalents. This map\ an extension of Cartan's spinor map [Ref. 4], maps the bispinor \\i to a constrained set of SL(2,C) x U(l) gauge potentials A^ and a complex scalar field p, where a = 0, 1, 2, 3 is a Lorentz index and K = 0, 1, 2, 3. Since the Lie algebra of SL(2, C) is regarded as the complexification of the Lie algebra of SU(2), the gauge potentials A? for j = 1, 2, 3 are complex, while the U(l) gauge potential A" is real. A^ and p satisfy the following constraint: Al^A KP I P I gap (1) where K is contracted using the SU(2) X U( 1 ) Killing metric and g^p is the space-time metric. With this constraint, the Dirac bispinor Lagrangian comes from the following Yang-Mills tensor Lagran- gian L, in the limit of a large Yang-Mills coupling constant g: ' The Cartan map [Refs. 1, 2] maps the bispinor f to a triplet of complex antisymmetric tensors FJ* (where j = 1,2,3) of Carmeli class G [Ref. 3). Such Vf can be expressed as Ff^= pCAJAf-AfA^ +iejkn,AgAP,) where p is a complex scalar and Afe for K = 0,1,2,3 are SL(2,C) X U(l) gauge potentials satisfying (1). (ejkn, for j, k, m = 1, 2, 3 is the permutation symbol.) L = -iRe[ASf A^p] + (D;p)(D"p)-!!|p + 2m|^ (2) where m denotes mass. D„ denotes the Yang-Mills covariant derivative with connection coefficients A^, and A^p is the Yang-Mills cur\ alure tensor associated with the gauge potentials \^. The indices are con- tracted using the Killing metric as well as the space- time metric. All bispinor observables can be derived from L using Yang-Mills formulas. .Although previous authors [Refs. 5. 2] derived the tensor form of the Dirac Lagrangian. they did not put it in the gauge symmetric Yang-Mills form (2). The Dirac equation can be derived from the Lagran- gian L by ascribing to the Yang-Mills field A^ a large self-couphng constant g. To be consistent with obser- vation, the fields A^ and p must couple more weakly (by the factor 1/g) with other fields. In particular, Ein- stein's equation becomes G^p = kT„p where G„p is the Einstein tensor, k is the gravitation constant, and T„p is the energy-momentum tensor derived from the La- grangian L. In the limit of large self-coupling g (ne- glecting terms in T„p not containing g) we have T„p = g T'ap where T'„p is exactly the usual Dirac energy- momentum tensor [Ref. 5]. Hence k' = kg is the ob- served gravitational constant, not k. Note also that in the Lagrangian L. the observed mass is m' = mg, not m. Then, as g tends to infinity, the Lagrangian kL is independent of g. In this limit, which we henceforth assume, we have: Lim k L = k' L' (3) a — 'X where L' is exactly equal to Dirac's bispinor Lagran- gian [Ref. 5]. Thus, as previously stated, Dirac's bis- pinor Lagrangian is a limiting case of the Yang-Mills Lagrangian (2), in which the self-coupling constant g tends to infinity. II. FERMI QUANTIZATION We quantize A^ and p by defining the classical Hamil- tonian to be: (Let SCR^ be a large cube.) 381 PRECEDING PAGE BLANK NOT FILMED H = Is T°°dx (4) where T"^ is the energy-momentum tensor derived from the fermion tensor Lagrangian (3). We maice a classical change of variables that simplifies H. The resulting Hamiltonian equations are then formulated as Heisenberg operator equations. Because the SL(2, C) X U(l) gauge group is not com- pact, H is not bounded from below. This has the con- sequence that any quantization of the fields A„ and p must obey the exclusion principle; otherwise fermions descend forever to lower energy states. By the Cartan map [Refs. 1, 5] the energy-momentum tensor has an expansion of the form: T"P(x,t) zz T^^ (X) a„a(t) (5) where the sum is over all pairs of fermion modes p and q, and where Tp^ (x) are fixed functions of x e S, and apq(t) are complex functions of time t satisfying apq = Hqp. The bimodal expansion (5) is irreducible because it cannot be expressed in tensor terms as a sum over products of single modes, as is the case with bosons. The Hamiltonian (4) can be written in terms of the amplitudes apq(t) as follows: H copapp (6) where (Op is the frequency of the mode p. Note that for simplicity, the amplitudes apq(t) are defined to be con- sistent with the hole theory. The classical Hamiltonian equations (which are equiv- alent to the constrained Euler-Lagrange equations for A^and p) are given by: da, dt = (ap,, H) (7) where the Poisson brackets { , } are defined for the classical amplitudes apq(t) as follows: { "pq' "p 'q') 1 l^pq' bp'q Ep'^ Opq') (8) where 8pq equals one if p = q and zero otherwise. Formulas (6), (7), and (8) are noncanonical tensor Hamiltonian equations which cannot be formulated as canonical equations in tensor terms. Nevertheless, they are easily quantized by replacing the classical amplitudes apq(t) with Heisenberg operators, denoted asapq(t), and the Poisson brackets (8) with (equal time) commutators [ , ] as follows: L"pq' "p'q'J pq' p'q P'q Pq' The Heisenberg equations become: dan dt = -i [apq, H] (9) (10) where H is the operator version of the Hamiltonian (6). To further simplify these equations, we attempt to factor apq(t) into a product of operators: apq(t) = c;(t) Cq(t) (11) where the dagger (t) signifies adjoint. Since they do not occur explicitly in the Hamiltonian H, the new oper- ators Cp(t) a priori could satisfy any relations consis- tent with the commutation relations (9). We exploit this arbitrariness in order to satisfy the exclusion prin- ciple, previously discussed. At time t we define: ^p ^q ' *'q^p (12) All other equal time anti-commutators of Cp(t) are de- fined to be zero. Formulas (11) and ( 1 2) are consistent with the commutation relations (9) as required. It is clear that equations (9), ( 1 0), ( 1 1 ), and ( 1 2). while derived from the tensor Hamiltonian equations, are equivalent to Fermi quantization via bispinors. Thus, the tensor Lagrangian (3) leads to Fermi quantization without the use of bispinors. Again, without the use of bispinors, we may extend the tensor Lagrangian (3) to include the electromagnetic field. Quantization is straight forward due to the fad that the interaction term is a function of the fermion amplitudes apq(t), as well as boson amplitudes bn(t). in. QUANTUM GRAVITY Spinor structure can be defined on a noncompact space-time manifold M by specifying, at each point xeM, a set of Pauli spin-half matrices OabCx) satisfying [Ref 3]: MB- ^PAB' ^ ap (13) Formula (13) has a topological as well as a metric consequence. The topological consequence of (13) is that M must be parallelizable [Ref. 6]. The metric consequence is that g„p is constrained as in formula ( 1 ). Since, for noncompact parallelizable space-times for- 382 mulas (1) and (13) are equivalent, spinor structure is nothing but an indirect way of constraining the metric g(,p on such space-times. However, the tensor fields A^ and p satisfying the constraint (1) are more general than (13), since they can be defined on general space- times. Formula (13) presents a dilemma [Ref 7] for quan- tizing both gravity and the Dirac field, since the def- inition of the Pauli matrices a^B- depends on the grav- itational field g„p. The problem is resolved by identifying the degrees of freedom in the constraint ( 1 ) as follows. Consider a fixed metric g„p on M and define Pauli matrices b^^. with respect to g„p. The metric g„p on M is expressed by: gap = gap + h„p (14) We also express the gauge potentials A„ by: A^ = ftA^ (15) where A^ satisfies the constraint ( 1 ) with respect to the fixed metric g„p. The dynamical fields are then A„ , p, and h„p provided that the matrix f = f^ can be uniquely solved as a function of h„p. Since A^ and p have bis- pinor coordinates with respect to the fixed spinor structure on M, the fields A^, p, and h„p can be quan- tized as in Section II. It remains to solve for the matrix f in formula (15) using the constraint (1). Since A?AKp = -|p|'gap (16) formulas (14) and (15) give: gya fl fp = gap + h„p (17) The solution of (17) is given by: f=Sci'2h" (18) n=0 where C|^ denote the binomial coefficients, and the matrix h is defined by: h = hP = t^ h,„ where g"P is the inverse matrix of g^p. (19) For the power series (18) to converge, the eigenvalues of h must lie within the unit circle. This restricts the validity of quantum gravity to small fluctuations of g„p about the fixed metric g^p. IV. CONCLUSIONS In this paper we have adhered to the program of first defining all fields, Bose and Fermi, as classical tensor fields, and then quantizing them using Hamilton equa- tions and Poisson brackets. From this vantage point, the Dirac equation becomes a classical tensor equation on the same level as the electromagnetic and gravita- tion tensor equations. Fermions, photons, and gravi- tons are obtained by quantizing the degrees of freedom allowed by the tensor constraint (1). We have shown in Section III that the constraint (1) implies that we can- not, in general, separate fermion and graviton degrees of freedom, except when the power series (18) con- verges. We also found that the fermion degrees of freedom require the use of noncanonical Hamilton equations (6), (7), and (8). Since the free Dirac tensor equation is completely integrable, we have shown that current us- age of only canonical Hamilton equations is too re- strictive for quantizing integrable tensor fields. REFERENCES 1. Reifler, F. and R. Morris, 1986, "A Gauge Symmet- ric Approach to Fierz Identities," J. Math. Phys., 27(11), pp. 2803-2806. 2. Zhelnorovich, V. A., 1990, "Complex Vector Trip- lets in the Spinor Theory in Minkowski Space," Proc. Acad, of Sciences of USSR, 311 (3), pp. 590-593. 3. CarmeH, M., Kh. Huleihil, and E. Leibowitz, 1989, "Gauge Fields," World Scientific, Singapore, pp. 21 and 40. 4. Cartan, E., 1981, The Theory of Spinors, Dover, New York, p. 41. 5. 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