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

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Frank Reifler and Randall Morris 

Government Electronic Systems Division 

General Electric Company 

Moorestown, NJ 08057 


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. 


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: 



I P I gap 


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|^ 


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. 


We quantize A^ and p by defining the classical Hamil- 
tonian to be: (Let SCR^ be a large cube.) 



H = 




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^^ (X) 



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: 




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: 



= (ap,, H) 


where the Poisson brackets { , } are defined for the 
classical amplitudes apq(t) as follows: 


"pq' "p 

'q') 1 l^pq' bp'q Ep'^ Opq') 


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: 


= -i [apq, H] 



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) 


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 


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). 


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]: 


^PAB' ^ ap 


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- 


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- 

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) 


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. 


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. 


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- 

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. 


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. 

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. Takahashi, Y., 1983, "The Fierz Identities— A Pas- 
sage Between Spinors and Tensors," J. Math. Phys., 
24(7), pp. 1783-1790. 

6. Geroch, R., 1968, "Spinor Structure of Space-times 
in General Relativity I," J. Math. Phys., 9 (11), pp. 

7. Ashtekar, A. and R. Geroch, 1974, "Quantum The- 
ory of Gravitation," Rep. Prog. Phys., 37, pp. 

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