The quantum state of a wormhole can be represented by a path integral over all asymptotically Euclidean four-geometries and all matter fields which have prescribed values, the arguments of the wave function, on a three-surface which divides the space time manifold into two disconnected parts. Minisuperspace models which consist of a homogeneous massless scalar field coupled to a Friedmann-Robertson-Walker space time are considered. Once the path integral over the lapse function is performed, the requirement that the space time be asymptotically Euclidean can be accomplished by fixing the asymptotic gravitational momentum in the remaining path integral. It is argued that there does not exist any wave function which corresponds to asymptotic field configurations such that the effective gravitational constant is negative in the asymptotic region. Then, the wormhole wave functions can be written as linear combinations of harmonic oscillator wave functions

Garay, Luis J.

English

NASA Technical Reports Server

v 93-27339
ON INDUCING FINITE DIMENSIONAL
PHYSICAL FIELD REPRESENTATIONS
FOR MASSLESS PARTICLES IN EVEN DIMENSIONS
Vineer Bhansali
Lyman Laboratory of Physics
Harvard University
Cambridge, MA 02138
Abstract
Assuming trivial action of Euclidean translations, the method of induced representations
is used to derive a correspondence between massless field representations transforming
under the full generalized even dimensional Lorentz group, and highest weight states of
the relevant little group. This gives a connection between 'helicity' and 'chirality' in all
dimensions. We also state restrictions on 'gauge independent' representations for physical
particles that this induction imposes.
1. Introduction
For d = 2n + 2, the generalized Lorentz group SO(2n + 1, 1) commutation relations are
[JAB, JCD] = i( SAC JBD + 5BDJAC -- 5ADJBc -- 5BC JAD ) (1.1)
with A, B = 0,... (d - 1) where 5 is the d dimensional Kronecker symbol (SAB = 1 if A =
B; 0otherwise), the boosts in the ith direction are defined as
Ki = iJio = -iJoi (1.2)
and for i,j _ O, the rotation generator in the i,j hyperplane is Jii = -Jji.
Alternatively, the commutation relations may be written
[JAB, JCD] = i(gACJBD + gBDJAc -- gADJBc -- gBCJAD) (1.3)
where g = diag(-1, 1,..., 1, 1) with the boosts generated by
I(i = Ji0 = -J0i (1.4)
and the rotations generated by Jo = -Jji, for i # j. The -1 in the metric in (1.3) arises
from the i on the boost generators in (1.1) ; to 'Wick' rotate the noncompact algebra to
the compact algebra of the special orthogonal algebra SO(2n + 2), or vice versa,
SO(2n + 1,1) _ SO(2n + 2)
iJio _ Ji,d (1.5)
g_5.
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S0(2n+2) - Dn+l hasrank = n+l, dimension = (n+l)(2n+l), #ofroots = 2n(n+l),
# of positive roots = n(n + 1), # of simple positive roots = (n + 1), where the roots
are all the raising and lowering operators in the algebra, the simple roots are the linearly
independent raising or lowering operators, and the positive simple roots are just the linearly
independent raising operators in the algebra.
The relevant little group of Wigner is defined to be the maximal subgroup of the Lorentz
group that leaves invariant the 'standard' momentum of a particle in d = 2n+2 dimensions.
We choose our massle._s particle to be moving in the d- 1 direction with d momentum
k s = (k,0,...,k);/t = 0,...,d-1. For d= 2n+2 the little group is generated by the
n commuting rotations J12, J34,..., J(2n-l,2n) which will be the Caftan generators (and
hereafter the 'weight notation', notated by a subscript W, will refer to the specification of
a state in terms of the eigenvalues of this ordered set of Caftan generators), and the 2n
'translations'
L + - Ji,2,+l +iJi,o, i = 1,...2n (1.6)
which, using (1.1) can be seen to form an Abelian subalgebra
[L+,L +1 =0. (1.7)
Each translation indexed by i is a sum of a boost in the ith direction and a rotation in the
i, d - 1 plane. The commutation relations of the little group are
[L,+,L+]-0
+]= i( ,kr$ - +) (1.8)
[Jij, Jkz] = i(6ikJjZ - permutations).
The little group (1.8) is not semi-simple (it has an Abellan subalgebra of translations) and
is isomorphic to E(d- 2), the Euclidean group in d - 2 dimensions.
L[ - Ji,2,_+1 - iJi,o -: L + - 2iJi,o (1.9)
also form an Abelian subalgebra (but they do not belong to the little group for above
choice of standard momentum), since
[L_,L-;] =0. (1.10)
Note also that
and under complex-conjugation
[L+,L-f] = 2iYo,i ¢ J
= 2J2n+a,o,i = j
(L,+)*= -L:
-), __ _L +.(L_
Much will be said about the significance of L + shortly.
(1.11)
(1.12)
282
We now state a key assumption: the translation generators annihilate physical
states. There are a number of reasons for this: (1) Finite dimensional representa-
tions only: The group transformation corresponding to the translation L +, is written
as D+(Xi) = e-ix'L+; the dimensionality of the representation is characterized by the
length of the translation vector _i )_' For finite dimensional representations the transla-
tion parameter Xi = O, Vi. (2) Gauge-independence: In general, an eigenstate of the Cartan
generators of the little group J12,... J(2_-l,2n) is not an eigenstate of the translation gen-
erators L + since they do not commute. The eigenstates of the Cartan generators can be
written as the d dimensional polarization vectors (e.g. in four dimensions the generator J12
which generates z-rotations has the eigenvectors d' = (0, 1, +i, 0)). It can be checked, for
instance, that the transversality condition required for the photon vector potential to be a
Cartan eigenstate is not invariant under finite translations. In fact, the translations gen-
erate effects identical to Abelian 'gauge' transformations [1] [2]. Thus the requirement of
trivial translations is the requirement that only 'gauge independent' objects be considered.
We believe that these two consequences of trivial translations are desirable from the point
of view of making a scattering theory for a finite number of physical degrees of freedom
without auxiliary conditions, as in [3] [4]. (3) 'Factorization' of invariant operators: It
can also be shown as a rather nontrivial consequence [5] that the eigenvalue of an invari-
ant operator in the enveloping algebra of the higher dimensional Lorentz group factorizes
into the eigenvalue of a generalized Pauli-Lubanski pseudovector and and a simple factor
related to the boost generator.
2. The Main Theorem
Only the main results are highlighted here and the interested reader is referred for fur-
ther details to [6]. Define a physical field A as a representation transforming under the
full higher dimensional Lorentz group and obeying the condition of trivial translations
L+A = 0, i = 1,2...2n.
Main Theorem: A physical field A i_ a highest weight of the Lorentz group.
Proofi Take the full group to be SO(2n + 1,1). The little group is then E(2n). By
definition, a physical field is annihilated by all the translations L+,... L+,. Now, a highest
weight is by definition the state annihilated by all the linearly independent raising opera-
tors. For SO(2n + 1, 1), which is rank n + 1, we need thus to find the n + 1 positive simple
roots and show that they all annihilate the physical field. To this end, we first want to
prove the following fact:
Lemma: All the raising operators can be made u_ing linear combinations of the translation
generators.
Proof of Lemma: Since there are 2n available translations, and n + 1 required linearly
independent raising operators, for n >_ 1, i.e. for four dimensions or more, there are cer-
tainly enough translations available to make all the linearly independent raising operators.
283
We choose the Cartan generators in SO(2n + 1, 1) to be J12,..., J2,,-a,2n-2, J2n+l,0. We
claim that the complete set of linearly independent raising operators for n > 1 is
Zl+± +
(2.1)
zL_l + iLL.
For instance, for the four dimensional Lorentz group (n = 1) the two linearly independent
raising operators are L + + iL +. To check, in any dimension that (2.1) is indeed the
complete set of linearly independent raising operators, commute each member of the set
with the Cartan generators using (1.1) and (1.8) to obtain the coordinates of the positive
simple roots in the Cartan basis. Translating this to the Dynkin basis [6] [7] obtain the
rows of the Cartan matrix. But since the rows of the Cartan matrix are defined to be
the coordinates of the linearly independent raising operators under commutation with the
Cartan generators, (2.1) is in fact all of them. With the physical field condition and (2.1)
the theorem is proved. QED
Note that using (1.11) we obtain
[L + + iL++I,L-[ - iL_.+l] = 4(J2n+1,0 + Ji,i+l), i ---- 1,3,5,2n- 1 (2.2)
and
[L+I - iL+,L-[ + ill] = 4(J12 - J30). (2.3)
Since the right hand side lies in the Cartan subalgebra, the lowering operator corresponding
to each of the raising operators is obtained by replacing L + _ L_-, i --* -i in (2.1). With
the definition
E + - A + - L + + iL +1,1,0,...,0 --
E + = A +
--1,1,0,...,0 -- _- L+I -iL+2 (2.4)
E_,_1,0,..., 0 = A- - L-i - iLl
E21,-1,o ..... o = A+ =- L-[ + iLf
where the subscripts on the roots E denote the eigenvalue under commutation with the
Cartan generators,
[A+,A+I = [A+,A+] = [A+,A-] = [At,A-] = 0 (2.5)
we note that A + and A +_ are raising operators in two orthogonal directions (with A- and
A+ the orthogonal lowering operators). In four dimensions, A_, A +_ are the two linearly
independent raising operators, and A-, A+ are the corresponding lowering operators. From
the physical field condition
L+A= L+A = 0,, A+A= A+A=0 (2.6)
284
which means that the physical field is a highest weight, and complex conjugating the last
two equations with the use of (1.12)
A-A* = A+A* = 0 (2.7)
which shows that the complex conjugate is the lowest weight. Looking, for instance, at
the weight diagram for the left and right handed spinors in four dimensions (in Dynkin
notation)
Left Right
(+1 0) (0 + 1)
(-1 o) (o - 1)
(2.8)
we confirm the result of (2.7) that indeed the spinors are inequivalent and self-conjugate
(i.e. the weight diagrams reflect to minus themselves).
3. Helicity-Chirality Correlation in Higher Dimensions
In one of his classic papers on 'Feynman Rules for Any Spin', Weinberg [8] proves that
the annihilation operator for a massless particle of helicity )_ and the creation opera-
tor for the antiparticle with helicity -A can only be used to form a field transforming as
U[A]¢n(x)U[A] -1 = _,m Dnm[A-1]¢m(Ax) under those representations [A,B] of S0(3, 1)
such that A = B - A 1. This restriction arises solely due to the non-semi-simplicity of the
little group for massless particles; in particular due to the requirement that the Euclidean
'translations' of the little group act trivially on the physical Hilbert space. As a direct con-
sequence of Weinberg's result it is observed that in four-dimensions a physical left-handed,
helicity -j particle can only correspond to a representation [j + n, n], (n is an integral
multiple of ½) whereas a physical right-handed, helicity j particle can only correspond to
the representation In, j + n]. The generalization of the statement to higher even dimensions
will be stated as the following corollary to the main theorem:
Helicity-Chirality Correlation: A physical field of the full group corresponds to a high-
est weight state of the little group (given trivial action of translations), and the eigenvalue
of each generator common to the full group and the little group remains unchanged under
the projection of a representation of the full group to a representation of the little group.
Proof." Since the little group with trivial translations is just the orthogonal group in
two lower dimensions, a subset of the linearly independent raising operators of the full
group is exactly the complete set of linearly independent raising operators of the little
group, with the caveat that they have to be appropriately Wick rotated to obtain the
compact form. For SO(2n + 1, 1) (n > 1) as the full group, the subset of (2.1) without
the last raising operator, L2n-1 + iL2n, upon Wick rotation, is the complete set of linearly
independent raising operators for SO(2n). For example, for S0(5, 1) the raising operators
are L + ±iL + , L + +iL +, whereas for the little group of trivial translations ,-- SO(4) they are
L + + L +. Since the full set of raising operators annihilates the full group highest weight,
1 A and B correspond to independent SU(2)_s.
285
and the projection from the full group to the little group is an orthogonal projection (since
the boost generator is the only non-common generator and it commutes with the Cartan
generators of the little group), it follows that the little group state is a highest weight state
of the little group, and most importantly, the eigenvalue of the Cartan generators of the
little group is invariant under the projection (hence the consequences of the corollary are
most explicit in the weight notation). To recapitulate, the heticity-chirality correlation in
higher dimensions is nothing but the fact that under projection of the field representation
from the full Lorentz group to a little group state, the eigenvalue of the Cartan generators
remains unchanged.
In conclusion, we demonstrate that our main theorem and corollaries reproduce some
familiar results of four dimensions:
.In weight notation, the last result shows why a chiral left-handed field transforming as
(__½,1 1_)w ([_,0] in conventional SU(2) x SU(2) notation) corresponds to a helicity -_
particle and a chiral right-handed field transforming as (½, ½)w ([0, !]2 in conventional
notation) corresponds to a helicity ½ particle.
,Using ½(J + iK) = A, B and the definition J3A - J12A = ,kA, with A the helicity, we get,
on using the physical field condition and (2.1)
(L+I -iL+)A = O _ (A1 - iA2)A =0 _ A3 = -A
(L + + iL+)A = O_ (B1 +iB2)A=0_Ba=B
(3.1)
since A, B generate independent SU(2) algebras and A1 - iA2 is the lowering operator for
one and B1 + iB_ is the raising operator for the other. Then, by definition Ja = As + B3 =
B - A = A which is Weinberg's condition [8] .
.As mentioned earlier, the translations are also generators of Abelian gauge transforma-
tions. Requiring them to be trivial restricts us to gauge independent, finite dimensional
repesentations of the full group. Our theorem and the corollary then tell us what is the
little group representation corresponding to this gauge independent full group represen-
tation. For example, the representation corresponding to the field strength tensor in the
conventional (SU(2) x SU(2)), Dynkin and weight basis is:
F _'" = [1,0] + [0,11 =(2,0)0 +(0,2)0 = (-1,1)w + (1, 1)w (3.2)
which has Jl 2 eigenvalues 4-1 and so is admissible as the J12 eigenvalue remains unchanged
and corresponds to the correct helicity of the photon as we project to the little group state.
However, the vector potential corresponds to
_ 1 (1,1)o (0,1)w (3.3)A" =[ ,_]= =
which has a J12 eigen,_-alue of 0 which does not correspond to a transversely polarized
helicity 4-1 photon.
I am grateful to Prof. Howard Georgi for his insightful comments, as well as for travel
support to participate in the Workshop on Harmonic Oscillators. I would also like to thank
Prof. Y.S. Kim and Dr. D. Han for inviting me to present this work.
286
References
[1] S. Weinberg, Phys. Rev. 135, B1049 (1964)
[2] D. Han, Y.S. Kim, Am. J. Phys. 49(4), 348(1981); D. Han, Y.S. Kim, D. Son, Phys.
Rev. D25, 461 (1982); D. Han, Y.S. Kim, D. Son, Phys. Rev. D31,328 (1981); D.Han,
Y.S. Kim, D. Son, Phys. Rev. D26, 3717 (1982)
[3] S. Coleman, B. Grossman, Nucl. Phys. B203, 205 (1982)
[4] V. nhansali, Mod. Phys. Lett. A5, 1223 (1990)
[5] V. Bhansali, in Classical and Quantum Systems: Foundations and Symmeteries Pro-
ceedings of Second International Wigner Symposium, Goslar, Germany (1991), World
Scientific.
[6] V. Bhansali, Intl. Jour. Mod. Phys. A7, 26 (1992)
[7] R. Slansky, Phys. Rep. 79, 1 1981)
[8] S. Weinberg, Phys. Rev. 134, B882 (1964)
287



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