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 quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors

Morris, Randall

Reifler, Frank

English

NASA Technical Reports Server

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.
I. CLASSICAL DERIVATION
We apply the double covering map from bispinors to
their tensor equivalents. This map 1, an extension of
Cartan's spinor map [Ref. 4], maps the bispinor tV to
a constrained set of SL(2,C) x U(1) gauge potentials A K
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(1) gauge potential Aoa is
real. A_ and p satisfy the following constraint:
A K AK_ = -Ipl2g,_ (1)
where K is contracted using the SU(2) × U(1) Killing
metric and g,_ 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:
I The Cartan map [Refs. 1, 2] maps the bispinor ¥ to a triplet of
complex antisymmetric tensors Fjal_ (where j = 1,2,3) of Carmeli
class G [Ref. 3]. Such jF_ can be expressed as l_j 1_= p(AoaA_-A_A_
+iaikrnA_A_ m) where p is a complex scalar and A_ for K = 0,1,2,3 are
SL(2,C) X U(1) gauge potentials satisfying (1). (ejk m for j, k, m = 1,
2, 3 is the permutation symbol.)
L = - _ Re [A_ A_ ] + (D-_) (D_p) - _ I p + 2ml 4
(2)
where m denotes mass. D_ denotes the Yang-Mills
covariant derivative with connection coefficients A_,
and AaK_is the Yang-Mills cur_ature tensor associated
with the gauge potentials A_. 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 K a large
self-coupling constant g. To be consistent with obser-
vation, the fields A_ and p must couple more weakly
(by the factor l/g) with other fields. In particular, Ein-
stein's equation becomes G,_ = kT,_l_ where Gap is the
Einstein tensor, k is the gravitation constant, and T,_
is the energy-momentum tensor derived from the La-
grangian L. In the limit of large self-coupling g (ne-
glecting terms in T,_O not containing g) we have T,_ =
g T'_I 3 where T',_I_ 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)
g---*_c
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 3 be a large cube.)
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H = _s T°° dx (4)
where T a_ is the energy-momentum tensor derived
from the fermion tensor Lagrangian (3). We make a
classical change of variables that simplifies H. The
resulting Hamiltonian equations are then formulated
as Heisenberg operator equations.
Because the SL(2, C) × U(1) 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 Carman map [Refs. 1, 5] the energy-momentum
tensor has an expansion of the form:
Ta[3(x,t) = Z Z Tpq_ (x)apq(t) (5)
P q
where the sum is over all pairs of fermion modes p and
q, and where Tpq_ (x) are fixed functions of x e S, and
apq(t) are complex functions of time t satisfying apq =
_qp. 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 = Z COpapp (6)
P
where COpis 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
AKand p) are given by:
dam = (apq, H) (7)
dt
where the Poisson brackets { , } are defined for the
classical amplitudes apq(t) as follows:
(apq, ap,q,) _--- -i (apq, 5p,q -- ap,q 5pq,) (8)
where 5pq 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
aS_pq(t), and the Poisson brackets (8) with (equal time)
commutators [, ] as follows:
[apq, ap,q,] : apq, 5p,q - ap,q 8pq, (9)
The Heisenberg equations become:
dapq _- -i [apq, HI (10)
dt
where I2Iis the operator version of the Hamiltonian (6).
To further simplify these equations, we attempt to
factor :_pq(t) into a product of operators:
apq(t) --_ _tp(t) Cq(t) (11)
where the dagger (t) signifies adjoint. Since they do not
occur explicitly in the Hamiltonian IZI, the new oper-
ators (_p(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:
_tp _q "_- CqCp^^'t = 5pq (12)
All other equal time anti-commutators of cp(t) are de-
fined to be zero. Formulas (11) and (12) are consistent
with the commutation relations (9) as required.
It is clear that equations (9), ( I0), ( I 1), and ( 12). 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 ofbispinors, we may extend the
tensor Lagrangian (3) to include the electromagnetic
field. Quantization is straight forward due to the fact
that the interaction term is a function of the fermion
amplitudes apq(t), as well as boson amplitudes b,(t).
III. 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 O_B,(X) satisfying
[Ref. 3]:
($_B,(Y 13AB' : g ctl3 (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_ 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 K
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 O_a, 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 _,,,p on M and define Pauli
matrices 6_,B, with respect to g,_l_.The metric g_13on M
is expressed by:
gc,p = _ap + hap
We also express the gauge potentials A K by:
(14)
(15)
where A K satisfies the constraint (1) with respect to the
fixed metric g_p. The dynamical fields are then A K, p,
and h,_p provided that the matrix f= _ can be uniquely
solved as a function of h_13. Since A K and p have bis-
pinor coordinates with respect to the fixed spinor
structure on M, the fields AsK, p, and h,_l_ can be quan-
tized as in Section II.
It remains to solve for the matrix f in formula (15)
using the constraint (1). Since
~K- 2Aa AKI_ = -Ipl L_
formulas (14) and (15) give:
(16)
g_ _ _ = gap + h_p (17)
The solution of (17) is given by:
f = Z C_/2 h" (18)
n=0
where C m denote the binomial coefficients, and the
matrix h is defined by:
h = h_ = _ h_a
where _P is the inverse matrix of _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 ofg_p
about the fixed metric _,,_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.
. 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.
. Carmeli, 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.
. Takahashi, Y., 1983, "The Fierz Identities--A Pas-
sage Between Spinors and Tensors," J. Math. Phys.,
24(7), pp. 1783-1790.
. Geroch, R., 1968, "Spinor Structure of Space-times
in General Relativity I," J. Math. Phys., 9 (11), pp.
1739-1744.
. Ashtekar, A. and R. Geroch, 1974, "Quantum The-
ory of Gravitation," Rep. Prog. Phys., 37, pp.
1211-1256.
• u.s c.,ow.._,,rMu_,'rpparcrtNc omct: _992.627.06 _, 60 : 3 383


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

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