Multiplicative anomaly and finite charge density

NUCLEAR PHYSICS

Multiplicative Antonio

A

Nuclear Physics A642 (1998) 222c-227~

ELSEVIER

Filippi”

anomaly

and finite charge density

??

“Theoretical Physics Group, Imperial College, Prince Consort Road, London SW7 2B’Z, United

Kingdom

When dealing with zeta-function regularized functional determinants of matrix valued differential operators, an additional term, overlooked until now and due to the multiplicative anomaly, may arise. The presence and physical relevance of this term is discussed in the case of a charged bosonic field at finite charge density and other possible applications are mentioned. 1. INTRODUCTION In field theory we often have to deal with functional determinants of differential operators. These, as formal products of infinite eigenvalues, are divergent objects (UV divergence) and a regularization scheme is therefore necessary. One of the most successful and powerful ones is the zeta-function regularization method [l-5]. It permits us to give a meaning to the ill defined quantity In det A, where A is a second order elliptic differential operator, through the zeta function (‘(s]A) = Tr A-“, which is well defined for a sufficiently large real part of s and can be analytically continued to a function meromorphic in all the plane and analytic at s = 0. As such its derivative to respect to s at zero is well defined and the logarithm of the zeta-function regularized functional determinant will then be defined by lndet $

= -C’(O]=l) - C(OIA) In M2,

(I)

where M2 is a renormalization scale mass. Sometimes, however, the differential operator takes a matrix form in the field space, as is the case with the two real components di of a complex scalar field. In this case we end up evaluating a quantity of the form lndet(AB), with A and’ B two commuting The fact is that it is not always true that the equality pseudo-differential operators. lndet(AB) = lndet(il) + lndet(B) holds. On the contrary, an additional term a(A, B), called the multiplicative anomaly (MA) [6-81, may be present on the right hand-side and eventually have physical relevance [9,10]. In this work I will introduce this quantity, compute it and analyse its physical relevance in the case of a charged scalar field at finite temperature and charge density, as well as present other possible physical systems in which it could play a role. *The author wishes to acknowledge financial support from the European Commission under TMR contract N. ERBFMBICT972020. 0375-9474/98/$19.000 1998 Elsevier Science B.V. All rights reserved. PI1 SO375-9474(98)00520-X

223~

A. Filippi / Nuclear Physics A642 (1998) 222c-22 7c

This and Evans 2.

work has been developed

L. Vanzo

and S. Zerbini,

for stimulating

in collaboration in Trento

The relativistic complex density has given rise to a In our recent paper [lo] overlooked until now, could avoiding the mathematical dimensions

The relevant for this system, =

where

H is the Hamiltonian

ature.

Aij is the elliptic,

2ie& In this

lnZp(p)

the system

and T.

DENSITY

has been

is the grand

case computing

of the system, non-self-adjoint,

studied

in generic

canonical

the partition

crucial

point

Q the charge matrix

-2ie$, __a; - V2 + ,s

minant and a functional one. one first [11,13-151. Stimulated procedure [181.

Now, the determinant:

(Spain).

D dimensions.

partition

function,

which,

=

- V2 + m2 - e2p2

(

in Barcellona

goes also to R. Rivers

scalar field at finite temperature in the presence of a net charge certain interest during recent years [ll-161. it is shown that, in a coherent regularized approach. the M,4, play a role in this system. I will try here to outline the results machinery. For clarity, I will mainly restrict myself to four

although

q+)

-8,

CHARGE

quantity for my proposes is [11,12,16]

,-W--Q)

Tr

with E. Elizalde, My thank

discussions.

THE BOSE GAS AT FINITE

space-time

(Italy).

- esl-ls

function

$

)

and p the inverse differential

of the temper-

operator (13)

.

requires

taking

both

an algebraic

deter-

The standard procedure consists in taking the algebraic by a recent criticism [17], we showed the validity of this

is that

we have

two possible

/I/I

= -ilndet

valued,

factorizations

= -ilndet

= -iIndet

for this

i . I 1

algebraic

s&

where:

I<* = -V2

+ m2 + (@

* iep)2

L* = -8, +

(d-V2

+ m2 f ep)2,

I will avoid here the standard steps that lead to the computation the partition function, as the reader will find them in greater detail For the K* factorization we obtain ln Zp(h;,

h’_)

=

&

[m*(ln $

of

- 3/2)]

J -ln(l-e-O(~-e~))_l./~ln(l_e-“(~+’,i), d3k

-v

of the logarithm in ref. [lo].

(5)

(6)

(2n)3

where the expected contributions for vacuum, particles and antiparticles are manifest. From now on I will represent the thermal contributions as S’(/~,/L). Similar manipulations can be done for the other factorization L*. In this case though, the chemical potential does not appear in the sum over Matsubara frequencies, but remains

224c

A. Filippi/Nuclear

with the momentumintegral

Physics A642 (1998) 222c-227~

and therefore

the term linear

in ,0 will be chemical

potential

dependent: In Zo(L+,

L_)

=

In this system

J!C 32+

[m4(ln $

the importance

- 3/2)]

+ g

(y

of the MA is therefore

manifest.

ln(L_L+), this two options give two different results ized partition function if the MA is disregarded. 3.

THE MULTIPLICATIVE The multiplicative

anomaly

ao(A, B) = lndet(AJ3)

- ez&7?)

+ S(P,p) .

Despite

for a properly

having

(7)

ln(ZC_K+)

zeta-function

=

regular-

ANOMALY [6-81 is defined

- lndet(A)

as

- lndet(B)

(8)

where the determinants of the two elliptic operators are defined by means of the zetafunction method. I recall that D are the space-time dimensions. In principle, it could be computed directly as difference of the involved quantities. In reality, actual calculations are very complicated even for simpler operators. We can fortunately resort to Wodzicki’s results for a remarkably neat recipe. For any classical pseudo-differential

e-ikzAeikz.

This

admits

operator

an asymptotic

A there exists a complete

expansion

for ]lc] +

symbol

A(z, lc) =

co,

where the coefficients (their number is infinite) fulfil the homogeneity property A,_j(z, tk) = k), for t > 0. The number a is called the order of A. Now, Wodzicki [6] proved

ta-jA,_j(s,

that for two invertible, self-adjoint, smooth compact manifold without

a(A B) =

>

elliptic, commuting, boundaries 1Vn:

pseudodifferential

operators

res[(1n(AbB-a))2] 2ab(a f b)

on a

(10)

= a(B, A),

where a > 0 and b > 0 are the orders of A and B, respectively. Here the quantity res(A) is the Wodzicki non-commutative residue. It can be computed easily using the homogeneous component

A_~(z,

Ic) of order

-D

All this can be straightforwardly A(x,

k)r<*

=

In

[

(

of the complete

applied

k2 + m2 - e2pz + i2epk,)

symbol,

to our operators.

- In (k2 + m2 - e2p2 - i2epk,)]’

Simply expanding (12) and performing the above integration commutative residue. Remembering (10) and that the order have the related MA as a4(h;,

I<_)

=

f!..Ke2p2(m2 - -)I e2p2. Y7r2 [

3

As an example:

(11) we obtain of our operators

.

(12)

the nonis 2, we

03)

A. Filippi/Nuclear

The same Finally,

225c

Physics A642 (I 998) 222c-22 7c

can be done for L*, obtaining another expression for a4(L+, L-). we obtain including this two results in (6) and (7) respectively,

lnZp(K+,K_)

= lnZp(L+,L_)

=

$

[m4(ln $

- 3/Z)]

+ SW, CL)

_&)I,

(14)

and the logarithm of the partition function turns to be the same for the two different approaches. Although consistent now, our result is remarkably different from the one in the literature where the MA was disregarded. The physical relevance of this additional term will be discussed in the next section. More generally, this term can be easily computed for any space-time dimension D and turns out to be always vanishing for odd D [9,10]. It has also been computed for the selfinteracting

field [13,14], but there

determinants 4.

PHYSICAL

density

1

where

p = &a’“z~~+r’)

= q P

ah

arise when dealing

involved

with the regularized

[lo].

physical relevance of the MA the crucial quantity is the effective as a function of the of external sources, which can be expressed

--&ln%(ili +pi

W,P,~) =

&3(P)

p=pV

operators

RELEVANCE

To investigate the potential in presence charge

many difficulties

of the complicated

+

e2p”2

ap

the later

is an implicit

and the mean

aln&dP)

+

$2

+

field x2 = Q2 as e2p2)52,

(15)

ap

(16)

t expression

for the chemical

potential

as a function

of p.

The physical states correspond to the minima of the effective potential. located in aF = X(72 - e2p2) = 0. We find therefore: 1) an unbroken phase, IC = 0, ep < m: ax 2) a symmetry condensation.

Fo=minF P where

breaking solution, z # 0, ep = fm, giving the relativistic Bose-Einstein For our system, explicitly, the unbroken and broken phase are respectively

=

Ev-&S(p,p)+~+&

=

-pv~

1

&V is the vacuum

It is possible are, give some

aww

- s

contribution,

to see, under inconsistencies

a detailed

e2p2(m2 -

e

y)]

(17)

I

w(m”_2q) [

(18)

and

generic

in the broken

D analysis, phase

[lo].

that

these

expressions,

We have to remember,

as they though,

A. Filippi /Nuclear Physics A642 (I 998) 222c-22 7c

226~

that

we worked

appears

until-now

with regularized

in the partition

function

multiplied

but “unrenormalized”

charge

by p, any ambiguity

density.

Since it

in it will correspond

to

an uncertainty in the free energy density of the kind PK. This constant h’ has to be fixed following physical requirements. A very reasonable one is that the symmetry is unbroken at T = 0, p = 0. For D = 4, K will be K = -$$-. For D = 4 only, this choice also removes the MA contribution to the charge the broken phase in any aspect and we get

density,

so that

the anomaly

does not alter

(21) (22) Also the critical tion. Is different

temperature (X = 0, ep = m) remains unchanged the unbroken phase, where the anomalous term

with this renormalizaremains:

,

(23) (24)

We should

now observe

at ultra rklativistic other hand, it does the broken

phase,

T N m. Notice, expression 5.

that

so that

it could

finally, that of the free energy

GENERAL

the anomalous

contribution

to the free energy

is non leading

temperatures T > m, since the thermal terms go as T4. On the not even contribute to the low temperature limit, corresponding to give relevant

corrections

the anomalous term is vanishing density for the uncharged boson

only in a intermediate

range

as e + 0, and the correct gas is recovered.

CONSIDERATIONS

This analysed is just one of the many possible physical systems were the MA could play a role [19]. The first question is therefore if this additional terms will always have physical relevance [li’,18]. This will in general depend on the system. As an example, for the above case the anomaly is vanishing for any odd dimension. In other cases [9] the anomaly could be simply non physical, as it could be reabsorbed in the renormalization procedure. Work is currently in progress on other systems, including fermionic ones. It is not difficult to see that for a single free fermionic field the anomaly is always vanishing, too. On the other side, it could play a role for neutrino mixing, in relation with recent results regarding inequivalent representations of the vacuum [20] and, in general, any time when there is a possible mixing or rotation in the functional space of the fields [15]. This needs further investigation due to the deep connection between the MA and the functional measure, which goes to the roots of the definition of the functional integral itself. The other relevant question is, of course, if the anomaly is regularization dependent [al]. Zeta function regularization is just one example of a wider class of regularizations called: “generalized proper-time regularizations” [22,23], f or which we showed the anomaly to be present [4,10]. This topic is also under further investigations and created a vivid debate

A. Filippi/Nuclear

22lc

Physics A642 (I 998) 222c-22 7c

lately. Here too, the answers are probably to be found in a proper and consistent definition of the ill-defined functional determinant itself, where this regularization approach ha,s. up to now, proved to be rigorous and coherent [24]. Note added in proof: After my talk. a work [25] by McKenzie-Smith and Toms appeared. There, the relevance of considering the MA within a functional integral approach is recognized although they do not agree on its physical relevance for the charged bosonic field.

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Yu.1. Manin, editor, M. Kontsevich and

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