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Covariant Quantization of Spinor Fields in a Given Gravitational Field.

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Annalen der Physik. 7. Folge, Band 41, Heft 3, 1984, S. 164-171
J. A. Barth, Leipzig
Covariant Quantization of Spinor Fields
in a Given Gravitational Field
By ECKHARD
KREISEL
Einstein-Laboratory, Academy of Sciences of GDR, Caputh
Prof. D. D. Iwanenko zum 80. Geburtstag gewidmet
Abstract. In coupling gravity with the quantum field theory, unitary transformations, depending on space-time-points, were considered and derivatives were introduced, which imply a nonintegrable parallel transport of the state vectors of Hilbert space [l].The Dirac equation, built with these
generalized derivatives, is quantized in a prescribed classical gravitational field. The quantization
can be performed in complete analogy to the usual procedure in Minkowski space, but the quantum
state vector becomes path dependent. In carrying out the quantization, two two-component classical
spinor fields necessarily occur, which obey Weyl’s equation. The considered quantized Dirac equations are also picture-covariant, that is they have the same from in each physical picture, especially
in the Heisenberg picture and the Schrodinger picture.
Kovariante Quantiiierung von Spinorfeldern im vorgegebenen Gravitationsfeld
Inhaltsubersicht. Zur Ankopplung der Gravitation an die Quantenfeldtheorie wurden ortsabhingige unitare Transformationen betrachtet und Ableitungen eingefuhrt, die einen nichtintegrablen Paralleltransport fiir Hilbert-Raumzustinde implizieren [l].Die mit diesen Ableitungen gebildete
Dirac-Gleichung wird in einem vorgegebenen klassischen Gravitationsfeld quantisiert. Die Quantisierung kann in voller Analogie zum ublichen Verfahren im Minkowski-Raum durchgefuhrt werden,
jedoch die Zustandsvektoren werden wegabhgngig. Bei der Durchfuhrung der Quantisierung treten
notwendig zwei klassische zweikomponentige Spinorfelder auf, die die Weylsche Gleichung edullen.
Die betrachteten, quantisierten Dirac-Gleichungen sind auch bildkovariant, d. h., sie haben in jedem
physikalischen Bild die gleiche Gestalt, speziell im Heisenbergbild und im Schrodingerbild.
1. Introduction
I n ref. [I] unitary transformations, depending on space-time-points, were considered
and derivatives were introduced, which imply a non-integrable transport of state vectors
of Hilbert space. The main argument was the fact, that the principle of equivalence and
the principle of general relativity give together with the corresponding classical field
theory only a prescription, how to couple the field operators of quantum field theory
with gravity. But these classical coupling prescriptions do not contain any assertion
about the influence of gravity on the physical states of the Hilbert space of specialrelativistic quantum field theory. Usually it is assumed, that one can work in Heisenberg picture, that is, one can cancel the possible influence of gravity on the states by
unitary transformations. But this would be the case only, if this influence is integrable.
In ref. [I] it was assumed, that similar to the geodesic parallel transport, the parallel
165
E. KREISEL,
Covariant Quantization of Spinor Fields
transport of states in Heisenberg picture becomes non-integrable, if gravity is switched
on. The equations for particle states with energy-momentum q in Heisenberg-picture
Here the derivative a, is an operator in Hilbert space, which is the identity times the
usual partial derivative with respect to a constant system Icn) of basic vectors of the
Hilbert space :
Generally it yields for the commutator:
[a,, avi 10)= 0.
(1.4)
The derivative 0,in (1.2) is non-integrable. Instead of (1.4) we get for the commutator:
[V,, Ovl I@) =
Here
w,rv- ayr,,
- [r,,rv11 I@)
*
0.
(1.5)
r, represents the assumed non-integrable action of gravity on the states I@).
I', is an operator in Hilbert space and was introduced in ref. [l]to behave like an affinity
under active unitary transformations, which occur in going over from one physical
picture to the other, for instance between Heisenberg picture and Schrodinger picture.
These active unitary transformations have nothing to do with changes of basic vectors
as used in (1.3).
Introducing I', as an affinity under the change of the physical picture, one has the
advantage, that one has to take then the derivative a, to be invariant under active unitary transformations. So for an active unitary transformation v^ it yields:
and the derivative in (1.2) goes over to,
v,
A
=
a,
-+ir,
h
with
F,= i?r,6+ - (a,u) u+,
A
h
(1.8)
(for details see ref. [ l ] ) .
The derivative (1.7) was used in ref. [l]for an alternative picture-covariant quantization of the generalized Klein-Gordon-equation.
+
g~vO,Ovp, may = 0,
(1.9)
where the second derivative in (1.9) also implied the general covariant Christoffel-term,
of the given classical gravitational field g,,,. The quantization could be performed in
complete analogy to the scalar field in Minkowski-space in a system of reference %
,:
distinguished by the covariant conditions :
(1.10)
166
Ann. Physik Leipzig 41 (1984) 3
where r;v
denotes the Christoffel-affinity built with g,,,. The main difference to the flat
case was only the equation for the parallel transport of particle states (1.2), which became non-integrable.
We shall now give a quantization for Dirac fields within the same frame work.
2. Picture-Covariant Quantization of Dirac-Fields
We consider the Dirac equation in a prescribed gravitational field gVy:
+
(yW,
m) Y = 0
(2.1)
and the prescription for parallel transport of tha stats vectors for particles with energymomentum q :
V,l@,>
= (8,
+ irp)l@/J
= 0.
(2.2)
V, acts on the field operator !P accordirig to:
V,Y = q,,+ qr,,Y].
The derivative
(2.3)
Here "!(" denotes the usual covariant derivative of a spinor field in a given gravitational
field. The yp-matrices are given by the constant Minkowskian matrices ya and a tetrad
i E compatible with the metric g,,, according to [2, 31 :
0
0
y, = i@
U Ya 3
where we use the following representation of the matrices y":
gpv = q,bh$ht,
(2.4)
(A,B = 1,2)
0 O-aAB
y.=-i(
4
iB o ) ,
( A , B = i, i)
with the hermite-symmetric Pauli matrices :
qab = a a A 4 s $ h ,
= O$.;i.
The indices p, a,A , A are raised and lowered by the Riemannian metric g,,,, the Minkowski metric qaband the antisymmetric spin nietric y AR , y~ B, respectively. Denoting
usual two-component spinors by xA, VA,the spinor field !P in (2.3) is represented by the
bi-spinor
05B
(2.6)
Now the covariant derivative for spinor fields is given by the following affinity [4]:
1
OVA B c,,RB;,
=2 Ra%:;,Cr;:
BObSB,
(2.7)
therefore equation (2.3) takes the form:
(2.8)
We see, that into the generalized Dirac equation (2.1) an additional term, the commutator of
and the field !Pis introduced by the derivative (2.8). This term is not present
in the usual frame-work of quantization in curved space, i t results from the concept,
discussed in the introduction. The resulting equation (2.1) for the field operator !P is
now generally covariant and, by the behavior (1.8) of rp,
picture-covariant too.
r,
E. KREISEL,
Covariant Quantization of Spinor Fields
167
In case of the Klein-Gordon field we could solve equation (1.9) by postulating translational equations and dispersional relations as they yield in Minkowski space :
w,,
v,v =
TI 9
(2.9)
(2.10)
g’”[H,, CH,, TI1 = m2.
Therefore we shall try a similar procedure for the equation (2.1) too. Such translational
equations must be integrable, to allow a solution of (2.1). That means, these equations
will contain a generalized concept of teleparallelism for the operators !P. This is realized
by introducing reference-spinors in two-dimensional spin-space. We consider the following translational equations :
(2.11)
D,Y
i[F,, Y ] = i [ H , , Y]
with
D,Y-( a d A
%xB .
(2.12)
-DBA,VB ’
a,vi
+
)+(
)
Here the new spinor affinity DAB, is assumed to be integrable, that means, there exist
two two-component linear independent spinor fields, which are covariantly constant
with respect to the derivative D,:
DPNtL) a,N&) D$,N$,) == 0
(L,
M = 1, 2).
(2.13)
+
We can represent D , by these fields, if we take them as orthonormalized according to:
(2.14)
(2.16)
(2.16)
In equation (2.11) H , is assumed to be connected with the energy-momentum-operator
of the Dirac field, therefore it is proportional to the identity with respect to spinor indices as
too.
The energy-momentum-operator H , must transform like an operator also under
active unitary transformations, because (2.11) should be a picture-covariant equation
iF, is a picture-covariant derivative already.
too, and D ,
The integrability condition of equation (2.11) results by differentiation; using (2.12):
r,
+
(2.17)
Now the curvature-tensor of D , ((2.16)) vanishes in (2.17) and the integrability condition (2.17) reduces to:
(2.18)
0.
i[& ( H , - r,)l- [(a, - r,),
(H, -
ru)l
Equation (2.18) has the same form as the condition for the integrability of the translational equations for the Klein-Gordon field in ref. [l].
Ann. Physik Leipzig 41 (1984)3
1G8
When equation (2.18) is fulfilled, then there exists a n active unitary transformation
U , yielding:
(2.19)
(a,U) U += i ( H , - F,).
A
Therefore by a certain active unitary transformation U , we can go over to a physical
picture, in which the transformed quantities g,and p, are equal:
0
H, =
,-.
r,.
(2.20)
The picture-covariant equations (2.11) then imply for the transformed field operator Y:
h
D,Y = 0.
(2.21)
This means with (2.6) by (2.12) and (2.13), that the field operator Y i s a linear combination of the reference-spinors N&) with constant coefficients dL),
b, i,:
(2.22)
We have used here, that the derivative D , of Y A B , y A B ,y ( L ) ( Metc.
) vanishes of course.
9 in the form (2.22) is the general solution of the translational equations (2.11) in
the special physical picture, fixed by (2.20).
Let us now look for the condition, that this solution of the translational equations
leads also to a solution of the Dirac equations (2.1). With (2.3) and (2.11) we can write
for (2.1):
(2.23)
(ypV, m ) Y = y@(Yll,- D,Y)
iy"[H,, Y ] m Y = 0.
+
+
+
Since equation (2.1) is picture-covariant, it yields also in the physical picture (2.20):
+
+
(2.24)
yJ'(Y,I,- D,Y)
iy"[H^,, Y] m Y = 0.
This is reduced by (2.21), which is generally fulfilled by the solution (2.22) of the translational equations (2.11) in the picture (2.20), to:
Y'YIlr
+ iy"2,, 31 + mY = 0.
(2.25)
The first part of the equation (2.25) can be understood as a condition for the spinors
NfL).With (2.22), (2.4), (2.5) and (2.7) the first term of (2.25) takes the form:
(2.26)
If we now postulate for the reference-spinors NtL, Weyl's equations:
O!ii~p,)I,, = 0,
(2.27)
then (2.26) vanishes identically, because the covariant constancy of IS%&and y A Band
the hermite-symmetry of 02g together with (2.27) also implies:
# p 4BN'Bllp
G
= O*
(2.28)
Therefore the Dirac equation takes within a system of reference spinors, distinguished
by (2.27), and a physical picture, given by (2.20), the form:
iy'[i?,,
$1
+ mY = 0.
(2.29)
Since according to (2.22) 9 is a linear combination with constant operators dL),b ( i ) ,
we can start to formulate within this picture the field operators 9
: (q) for particles and
antiparticles with given energy-momentum q and spin S.
E.KREISEL,Covariant Quantization of Spinor Fields
169
To fulfil the Dirac equation (2.29), we have to introduce the creation and annihilation operators in such a way, that yields:
[&
@$@)I
= &qp@$(d
(2.30)
*
Therefore we write the field operators p : ( q ) with annihilation operators for particles
A 8 ( q )and creation operators for antiparticles BJ(q)and the spinor amplitudes w$(q)
in the form:
h
y$ (q) = w$ ( q )A 8 ( q ) ,
@;
(!
=
I
w)
i-(q) B,t(q),
(8
(2.31)
= 1, 2 ) ,
where the usual anti-commutation relations of Fermi particles yield for A8(q),B,(q):
(2.32)
Then, introducing the enerw-momentum operator f i B by
’
/
l
= /qp(”)
( A’A!(q) A S ( q )
+ B:(q)
B8(q))
‘q
(2.33)
equation (2.30) is fulfilled because of (2.31) and (2.32). Using (2.30) and (2.31) equation
(2.29) can be written as follows:
(2.34)
fiq,,y”w$
mw$ = 0.
+
For later purposes it is more convenient, to write (2.34) in two-component spinors.
According to equation (2.22) we write for w$ with constant u $ ( ~ %&):
),
(2.35)
(2.36)
A
Multiplying (2.36) with NLM),respectively N($) and using (2.14), the resulting equations
take the form:
+
& q p i g f P ( ~ ) ( 4 Y $ z ) m y f )= 0 ,
+
(2.37)
&qpi%7?L)(p)usi(L) m%&f) = 0,
where
(2.38)
are the components of the Pauli matrices with respect to the reference spinors N&),
NA .
(L)-
Now the spinors N&), N$) will depend because of (2.27) in general on the spacetime coordinates xu.Therefore the matrices (2.38) in (2.37) are not constant. But they
Ann. Physik Leipzig 41 (1984)3
170
can be transformed back t o the constant Pauli matrices also by a certain local Lorentz
transformation wf(x”) acting on the Lorentz-index “a” of the matrices qL)(&):
qh(&)=.wf&ni)
(2.39)
= 4i6(,)
A 61’.
(M)‘
We need this transformation, to get really solutions t$(,),
@&) of (2.37), which are
constants, as they have to be in correspondence with (2.35). Therefore we have t o
choose the up t o now free q,(z”)in (2.37), (2.33) in the following way:
q,(x”)= wk”,n,,
(2.40)
where wg can be eliminated from (2.39) using the basic relations of the Pauli matrices
and (2.38):
wbr - ~ bA . Bc ~. ND (DLD)ND.
(MI? LA
YMi.
(2.41)
Then qa in (2.40) is the constant energy-momentum vector, measured in the tetradframe
-
h,a = wt(x”)hb,.
(2.42)
The energy-momentum vector qa has to fulfil the dispersional relation
qabqaqb- m2 = qi - q2 - m2 = 0.
(2.43)
Inserting (2.40) into (2.37) we get finally the usual Minkowskian Dirac equations in
momentum space :
(2.44)
where qo is
(2.45)
@&,(q) of (2.44) are well known and the
The linear independent solutions v$(A)(q),
general field operator Y,containing particles and antiparticles with all possible fourmomenta, can be written with (2.31) and (2.35):
This is the solution for the field operator in the physical picture (2.20). The general
expression for the solution Y of (2.1), is given by (2.46) and an arbitary unitary transformation V(x”,A,(q), A $ ( ( ) , B,(q),BJ,(q’))according to :
Y = VYVt,
v+= v-1.
(2.47)
In the special case, that g,” is Minkowskian, we have
w $ = sg,
N&) = sfl,,,
A;
=
s.,
(2.48)
and the unitary transformation V , leading to the general form of the solution of (2.1),
is just:
v = eiH,x,
(2.49)
E. KREISEL,Covariant Quantization of Spinor Fields
171
where the transformed operators &q), @ ( q ) because of (2.32), (2.33) und (2.48) are
(2.50)
Therefore the unitary transformation (2.49) just reproduces the common exponential
factor of the field operator.
One can easily verify, that Wejd’s equation for the spinors (2.27) implies the covariant condition (1.10) for the tetrads, used in ref. [ l ] for the quantization of the scalar
field. With the covariant constancy of o$h one can write for (2.27), transvecting it with
N&,
:
(4
kN&)),,,N&)
(2.51)
= 09
or equivalently :
(2.52)
(o/lAkN&)N&))llfi- A B. N (AL NB.
) (M)llp = O ,
where the second term vanishes because of (2.27). For the first term we can write with
(2.35) and (2.38):
(2.53)
(qL)(k))llP= ( W : b b ) l l a = 0 *
Therefore with (2.52) and the fundamental relations of the Pauli-matrices i t follows:
-
(2.54)
hi;@= (w:ii;.g);, = 0.
This means, that the quantization of Dirac-fields and of scalar fields can be consistently
performed within the same frame of reference.
The equation (2.2) for the transport of the state-vectors
introduced by the action
of the creation operators on the vacuum state lo), defined by:
(2.55)
’A,(q)lO)
= B,(q)lO) = 0,
,)@I
are again non-integrable a s in ref. [l]. The general remarks, made there in connection
with this problem, apply also for the states I@).
References
[l] KEEISEL,
E.: Ann. Physik (Leipz.) 39 (1982) 371.
D.: Nachr. d. Akad. d. Wiss. d. UdSSR 73 (1929).
[2] IVANENKO,
[3] FOCK,
V.; IVANENKO,
D.: Phys. Zs. 30 (1929) 646.
L.; VAN D. WAERDEN,
B. L.: Berlin: Sitzungsberichte der PreuB. Akad. u. Wiss. 1933,
[4] INFELD,
S. 380.
Bei der Redaktion eingegangen am 15. Mai 1984.
Anschr. d. Verf. : Prof. Dr. sc. EC-D
KREISEL
Einstein-Laboratoriumder AdW der DDR
DDR-1502 Potsdam-Babelsberg
Rosa-Luxemburg-Str.17 a
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