Universal Dirac-Yang-Mills theory

Volume 244, number 2

PHYSICS LETTERS B

19 July 1990

Universal Dirac-Yang-Mills theory G. Ferretti J and S.G. Rajeev 2 Department of Physics andAstronomy, University of Rochester, Rochester. NY14627, USA Received 18 February 1990; revised manuscript received 1 May 1990

An algebraic analogue of Yang-Mills theory coupled to fermions is studied. The pure gauge sector &the theory is the "universal Yang-Mills theory" proposed in an earlier paper. When coupled to fermions, it has the global (ehiral symmetries) and discrete symmetries (P and CP) of QCD. The case of one flavor is studied in detail in the presence ofa regularization. The most general solution of the Gauss' law constraint is found. It is shown that this theory is equivalent to a ( 1+ 1)-dimensional Fermi model.

An algebraic analogue o f Y a n g - M i l l s theory [ 1-3 ] incorporating fermions, called universal D i r a c - Y a n g Mills theory, is presented. A similar approach, where no fermions are present, has been discussed in refs. [4,5]; see also refs. [6,7] for further discussions o f this theory. These investigations can be viewed as a continuation o f the classic study o f the large N limit in refs. [ 8,9 ]. T h e y are also related to Connes' work on n o n - c o m m u t a t i v e geometry [ 10] in a way exp l a i n e d in refs. [4,5,7]. There might be also some connection with m a t r i x models for t w o - d i m e n s i o n a l gravity [ 11 ]. We will show that i f all the fermions transform under the fundamental representation, there are no nontrivial solutions. This is interesting because the corresponding Y a n g - M i l l s theory is anomalous. Instead we will consider the case where there are equal n u m b e r of fermions transforming u n d e r the f u n d a m e n t a l and conjugate fundamental representation. This is the universal analogue o f QCD. In fact the theory has the same chiral [ S U ( N f ) × S U ( N f ) × U ( 1 ) A ] a n d discrete ( P a n d C P ) s y m m e t r i e s as QCD. Universal gauge theory has the same anomaly structureas Q C D , so that the 't H o o f t a n o m a l y arguments would i m p l y that the chiral s y m m e t r y is spontaneously b r o k e n in universal D i r a c - Y a n g - M i l l s theory at least for N f > 2. ~r This work is supported in part by the US Department of Energy Contract No. DE-AC02-76ER13065. 1 BITNET address: FERRETTI@UORHEP 2 BITNET address: RAJEEV@UORHEP

We will study in detail the simplest case N f = 1. The constraint equation is solved completely in this case. It is shown that the theory is equivalent to a two-dimensional F e r m i gas. In an o r d i n a r y Y a n g - M i l l s theory, the h a m i l t o n i a n f o r m a l i s m in the t e m p o r a l gauge leads to the following hamiltonian: H = ~ d3x { ½ t r ( E 2 + B 2) - :q } ( k r . D ) q s : - : # } ( k r . D ) # f : },

( 1)

where f i s the flavor index and the fermions are ass u m e d to be massless. (We are using the chiral representation for Dirac matrices. ) In the Schr6dinger representation, the wave function must satisfy the constraint ~(gAg-~+gdg

-~, gqf, g*c]f) = ~ ( A , qf, t]f),

(2)

where g is an element o f the c o m p o n e n t o f the gauge group connected to the i d e n t i t y g:. ~ 3 ~ SU (3). The p r o b l e m o f solving Y a n g - M i l l s theory in the canonical f o r m a l i s m is to find the eigenstates o f the hamilt o n i a n H satisfying the constraint. Even the solution o f the constraint has so far been intractable. First o f all, we will describe the canonical formulation o f universal D i r a c - Y a n g - M i l l s theory. The starting p o i n t in constructing the theory is the extension o f the gauge group to the universal group U2. A detailed description o f U2 can be found in refs. [ 4,5,10 ]. F o r our purposes the construction o f the

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universal gauge group can be summarized as follows. Consider an infinite-dimensional Hilbert space and decompose it into the direct sum of two infinite-dimensional subspaces ~¢= ~_ e ~ + .

(3)

Define an anti-hermitian operator e with respect to this splitting:

,(o i 0i)

,,,

(Note: The convention in ref. [4] was that e is hermitian. ) Let U ( ~ ) be the set of all unitary operators on ~ . The universal gauge group Ue is defined tO be the subgroup of U ( J r ) made of all ~ such that Tr I [e,g]41 < o o .

(5)

The algebraic analogue of the Yang-Mills potential A and the electric field E are two anti-hermitian operators A and/~ defined as

X~

(It2 14) ff,~( I2 14/3"~ 14

Iz

'

\L/3

Ie]'

(6)

with respect to the splitting (3). Here Iv denotes the Schatten ideal [ 12 ] Ip ={X, Tr IXlP
(7)

The magnetic field /7, algebraic analogue of the curvature, is described by the hermitian operator /7--_[e,.~l++~e--(q~2+l) e ,

qb---e+A.

(8)

The operators E, B and • transform under the adjoint action of the universal gauge group. The analogues of the covariant derivative for E and B are [ q~,/~] and [ q~,/7] respectively. On the other hand, for two elements q and qtransforming under the fundamental and conjugate fundamental representation of U2 the analogues of the covariam derivatives are q~q and qb*q. All these facts lead quite naturally to the hamiltonian

(

/~=½ T r ( - / ~ 2 + / T z ) - i

-i ~ [q},e]~b~s). f=t

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19 July 1990

The presence of a minus sign in front of/~2 is due to the fact that /~ IS an anti-hermitian operator and therefore /~2 is a negative operator. The commutators for the fermionic variables are needed in order to preserve the discrete symmetries [see eqs. ( 11 ) and (12) below ]. The pairs (q, q~) and (q, 4t) satisfy the canonical anti-commutation relations

[q~,q~,]+=6~rfy,,

[q~, q~, ] + = 6 ~ 5 , .

The i n d i c e s f a n d f ' run over the flavors 1, 2 ..... Nf We have been forced to introduce for each variable qy a conjugate variable qy. It will be shown later that its presence is required in order to have non-trivial so, lutions for the constraint. The variables q and q can be thought as left-handed and right-handed components of the fermion. One may then regard the discrete symmetry transformations as

P: q~(l,

q~q,

q)~*,

(11)

and CP: q ~ q t ,

gl-'gf,

tI)--,-q).

It is also clear that we have global symmetry under

q~el°~hlq,

@~ei~h2~],

qb_~.

(13)

This is the analogue of chiral symmetry. The only global symmetry of QCD that is missing is the U ( 1 )v symmetry which is here part of the gauge group. So our theory is really analogous to QCD with gauge group U(No) rather than SU(Nc). But it is well known that this does not make a real difference in the dynamics of the theory. We will now study the case Nf= 1 in greater detail. There will be divergences in this theory analogous to those of QCD. Therefore, let us regularize the problem by assuming Yg=C N, with N even integer. The hamiltonian in the Schr~Sdinger representation follows from the substitution of the kinetic term _/~2 in eq. (9) by the laplacian

02 0q5 3qb "

The hamiltonian is therefore (9)

(12)

( h i , h2, ei") sSU(Nr) × SU (Nf) X U ( 1 )a:

_/~2~ ~ [q},q~]CI){ f=l

(10)

(14)

Volume 244. number 2 1Y [ 02 / 1 = 5 ~Tr ~ 0 ~

PHYSICS LETTERSB ) + ((bE+ 1) 2

- i [ q ~ , qk] (bJk--i [#~, ~k] q)~:],

(15)

to be solved together with the constraint for the wave function

T(~qb~- 1, ~q, ~,#) = T( qb, q, # ) .

( 16 )

First, let us construct the most general solution of the constraint. What was too hard a problem in ordinary Yang-Mills theory here turns out to be a fairly straightforward application of group theory. Let us first expand T i n terms of the Grassmann variables q and #, --Pl -/TN ~(qb, q, ¢) = ~ ~ue(qO)q /"1 1 ...qNI/Nq, ...qN ,

(17)

where v={vi}, ~ = { ~ } and u~, #~ take values zero or one. ~P(q~) can be thought as a 2Nx 2 N matrix of matrix elements ~P,~(4) in a fermionic Fock space. Note that if all the fermions transformed under the fundamental representation (so that there would be no ~ in the problem), there would be no solution that depends on q at all (to see this, consider ~ = e i" I). That is why we consider a theory with fermions in both fundamental and conjugate fundamental representations. Recall thai QCD would suffer from anOmalies if all the quarks transformed under the fundamental representation. In nature quarks come in pairs transforming Under conjugate representations of color. The same argument implies that the only non-vanishing matrix elements are those for which N

N

v~= Z ~ = k . i=1

(18)

j=l

In terms of these matrix elements the constraint reads ~J~,,(~,~-l)=A:,.v(rg)~(~)A~,~,(~-~)

,

(19)

where A (~) is a representation o f ~ on the fermionic Fock space. We exploit this fact and we choose ~ U (N) such that ~qb~- ~is diagonal. Note that ~ is ambiguous up to a left multiplication by an element in [U( 1 ) ]Nx SN. In particular, c h o o s i n g / ~ [ U ( 1 ) ]~, eq. ( 19 ) implies ~/tT,v ( * d i a g ) = U v ( ~ I ,

"-,

~)N)(~v,v,

(20)

19 July 1990

where u~ is some scalar function. Note that the statement that ~Pevaluated on a diagonal matrix q~aiagis itself diagonal in the fermionic Fock space is a nontrivial one and holds only by virtue of the constraint. In the following, we will indicate by z/the diagonal m a t r i x ~(~diag) for notational convenience. We still need to fix theSN part of the gauge group. For z:~ SN, the action of A (z0 is that of shuffling the indices of the matrix ~. Using this fact one can pick g~ SN such that v~(i) -=0= 1 for i ~ k , =0

for i > k ,

(21)

and one may define a single function V ( ¢ I . . . . , 0 N ) ~ U 0 ( O l , "'" OU)

= u,(~b~-,(i)) .

(22)

The remaining Sk × SN_ k symmetry reflects the fact that the function v (#t, ..., ~N) must be independently symmetric in the sets of variables (O~, ..., ~k) and (~k+ b ..., ON)- This exhausts all of the constraint. To summarize, one can find the most general solution of (16) in four steps: ( 1 ) Pick an integer k such that 0 ~
(Ok+,,..., 0~)(3) For a given set of occupation numbers { v~} satisfying (18) find ff as in (21 ) and write z/~ = v(0~-,(i))~., •

(23)

~ , -.., #N being the diagonal element of - iqbdi,g. (4) For a given anti-hermitian matrix q~, find a that will diagonalize it and write ~:.~( qb ) =A:,,, ( ~ - I ) @ . , , A,, , ( g -~ ) .

(24)

We can now substitute the solution for the constraint into eq. (15) and find the SchrSdinger equation for the function v. The SchrSdinger equation for v is most easily derived by using a variational principle. Let E[~]=½

d~trtr

0qb 0qb

+½ j dqbtr(1 +q~2)2~r ~t~p,

(25)

be the energy functional. The term in the hamilto267

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nian that couples q to q~ is zero on states that satisfy the constraint. However, q and qb still do not decouple because the constraint itself mixes them. The symbol ~r represents the trace in the Fock space and tr is the trace in the original Hilbert space C N, Actually, we will consider only the trace in the subspace given by Y~il,i= kbecause equations with different k's decouple. In eq. (25) we make the standard change of variables from the "cartesian" co-ordinates q~i to the "polar" co-ordinates ~ and 0~; ~z being the eigenvalue of - i @ and 0u the angular variable characteriZing the unitary transformation g (0 u) that diagonalizes q). The energy functional expressed in terms of the new co-ordinates becomes E[~]=½ fdCdO d ~

X

OqJi a~ ~ +gU~ tr O0u O0~

19 July 1990

1

+2

+ E (¢i-1)21vl2], i

(29)

/

where the symbol Am, represents the anti-symmetrizatlon of the function v in the variables 0,~ and ¢,. Going from eq. (25) to eq. (29) requires some work; we are planning to include the proof in a longer paper. We can now eliminate the 7 2 dependence by making the following transformation: Z=-~v. The coefficient 7 is totally anti-symmetric in its variables, Hence, the new wave function Z is anti-symmetric in the first k and in the last N - k variables. This substitution allows us to separate the kinetic part of the energy in a sum of N decoupled components because of the property [ 5 ] 027

+ ~/((~i=l)2tr ~t,~),

where gu~ is the metric induced by the change of variables. To be more specific, ds2=Y.kd¢2+ where COmn=tr(em, rg-1 E , , ~ , ( O n - ~ . ) 2mm.O)m,, * × dg) and era, are the Weyl matrices. We can use the constraint to relate the angular derivative of ~ t o the commutator of the wave function t~ with the generators A,,. of the U ( N ) action on the fermionic Fock space. After some non-trivial calculations, we can show _ oq, t o q , gU"tr

v

i~{[A_..°, ~]*[~m.,

O0 'u O0 v -- m/.~n

~]}

(~m--On) 2

= E ~"~'' (PlA~mnll)t)[bll"--Uu[2 m~.

(0m--¢.)2

(27)

"

Therefore the only angular dependence in the integral comes from det g and it can be integrated over, yielding the result [ 13 ]

f The energy functional c a n be rewritten in terms of the function v as 268

(30)

(26) The expression for the energy functional in terms of this new function Z is therefore 1

0

2

+2m.n'Om--1 ¢.)~z*Smnz ] .

'31'

The anti-symmetrization operator Amn has now been replaced by the symmetrization operator Stun acting on Z. Note that, in the last equation, we can extend the sum over m<.k
Volume 244, number 2

PHYSICS LETTERS B

02

19 July 1990

We t h a n k J. Mickelsson a n d R. M a r s h a k for discussions. 1

"~ m<~k
(32)

In the language o f second quantization, we can introduce the fields bl ( 0 ) a n d b2(0) for the two types o f fermions and write the second qnantized hamiltonian as 02

H=½ ~ dOb~(--~-~ +(O 2- 1) 2) ba l + f do dO' ( 0 z 0 ' )

2

X [Jll(O)J22(O')-J12(O)J12(O')]

,

(33)

the currents being defined as follows: Ja,, ( 0 ) = b*,(0)ba, ( 0 ) •

(34)

N o t e that eq. ( 3 3 ) displays an SU ( 2 ) s y m m e t r y that was not manifest in the previous expressions. The theory without the q-variables tends to a free ( 1 + 1 ) - d i m e n s i o n a l Weyl fermion [5 ] in the limit N ~ o o . We conjecture that the present theory will tend to a Thirring m o d e l with S U ( 2 ) s y m m e t r y when the regularization is removed. The solution o f this theory will illuminate the b e h a v i o u r o f the 0-vacua o f gauge theory with fermions. Also, we hope to study the case o f several flavors an a later publication. The a i m is to see whether chiral symmetry is spontaneously broken.

References [ 1] W. Marciano and H. Pagels, Phys. Rep. 36 (1978) 137. [2] G. "t Hooft, Nucl. Phys. B 72 (1974) 461. [3] E. Witten, Nucl. Phys. B 160 (1979) 57. [4] S.G. Rajeev, Phys. Lett. B 209 (1988); Universal YangMills Theory, in: Proc. Eleventh MRST Meeting, eds. C. Rosenzweig and K.C. Wali (Syracuse University, 1989). [ 5 ] S.G. Rajeev, Universal gaugetheory, University of Rochester preprint (1990). [ 6] M. Tanaka and K. Fujii, Algebraic analogue of Yang-MillsHiggs theory, preprint CPT#1365 (1988). [7] J. Mickelsson, Curent algebras and groups (Plenum, New York, 1989). [8] E. Brezin, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59 (I978) 35. [9 ] M.L. Metha, Random matrices and the statistical theory of energy levels (Academic Press, New York, 1967). [10] A. Connes, Pub. Math. IHES 62 (1985) 257; Commun. Math. Phys. 117 (1988) 673. [11 ] D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127; M.R. Douglas and S.H. Shenker, Nucl. Phys. B 335 (1990) 635; E. Br6zin, M.R. Douglas, V. Kazakov and S.H. Shenker, Phys. Lett. B 237 (1990) 43. [ 12 ] B. Simon, Trace ideals and their applications (Cambridge U.P., Cambridge, 1979). [ 13 ] H. Weyl, The structure and representation of continuous groups (Princeton, N J, 1955).

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