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Partition function for the eigenvalues of the

arXiv:hep-ph/9302233v1 8 Feb 1993BNL–GP–1/93January, 1993Partition function for the eigenvalues of theWilson lineAndreas Gocksch&Robert D. PisarskiDepartment of PhysicsBrookhaven National LaboratoryUpton, New York 11973AbstractIn a gauge theory at nonzero temperature the eigenvalues of the Wil-son line form a set of gauge invariant observables. By constructing thecorresponding partition function for the phases of these eigenvalues, weprove that the trivial vacuum, where the phases vanish, is a minimum ofthe free energy.

In computing the properties of gauge theories at a nonzero temperature T, a stan-dard method of calculation is to use the imaginary time formalism, where the gaugefields Aµ are strictly periodic in a euclidean time τ, with period β = 1/T [1]. Thisallows for a type of Aharonov-Bohm effect at nonzero temperature, characterized bya nonzero expectation value for the Wilson line that wraps around in the direction ofthe imaginary time, exp(igR A0 dτ).

At one loop order [2-4] these inequivalent vacuaare parametrized simply by a constant value for the gauge potential A0; calculationshows that the trivial vacuum, with A0 = 0, minimizes the free energy. At two looporder [5] it is necessary to chararcterize the vacua not just by constant A0, but bythe Wilson line itself [6,7]; after doing so, the trivial vacuum remains a minimum.As conjectured in ref.

[7], in this paper we show that the trivial vacuum is alwaysa minimum of the free energy. Our starting point is a gauge invariant quantity ineuclidean spacetime, the effective potential for the phases of the eigenvalues of theWilson line.

This euclidean path integral is then transformed into a partition function[8], where the constraint on the phases of the eigenvalues becomes an imaginary chem-ical potential for global color charge. Because this chemical potential is imaginary,it follows directly that the trival vacuum minimizes the free energy, up to standarddegeneracies.Typically the partition function of a gauge theory is a sum not over all states,but only over those which are gauge invariant [1,6,9].

This is true, for example, incomputing the free energy at nonzero temperature. In our case, while the sum overstates is gauge invariant, the individual states which contribute are not.

In this vein,we note that while partition functions with a chemical potential for global chargehave been studied previously [10], hitherto it was assumed that only gauge invariantstates contribute. The extension of what is an allowed partition function is, we feel,the most striking feature of our results.For definiteness we consider an SU(N) gauge theory without matter fields, fol-lowing previous conventions [7].

Under a local gauge transformation Ω, the gaugepotential transforms as AΩµ = Ω†DµΩ/(−ig), with g the coupling constant, andDµ = ∂µ −igAµ the covariant derivative. The Wilson line in the direction of eu-clidean time isL(x) = P exp igZ β0A0(x, τ) dτ!≡U(x) Λ(x) U†(x) ;(1)P denotes path ordering, and x is the coordinate for three spatial dimensions.

The1

matrix L(x) is unitary, and so we can write it as the unitary transformation of adiagonal matrix, Λ(x): Λij(x) = δijΛi(x), UU† = 1. Under a gauge transformationthe Wilson line transforms asL(x) →Ω(x, β) P exp igZ β0AΩ0 (x, τ) dτ!Ω†(x, 0) .

(2)Without loss of generality we require that both the gauge fields and the gauge trans-formations are strictly periodic in the euclidean time. If Ω(x, β) = Ω(x, 0), then forthe Wilson line a gauge transformation is just a similarity transformation; the matri-ces U(x) are gauge variant, U(x) →Ω(x, 0)U(x), while the diagonal matrix Λ(x) isgauge invariant.Notice that what we call the Wilson line is a matrix in color space.

This is is dis-tinct from what is usually termed Wilson (or Polyakov) loop at nonzero temperature,which is the trace of this matrix [1]. Since tr L = tr Λ, the trace of the Wilson line isautomatically gauge invariant.The phases of the eigenvalues are given by Λ(x) = eiλ(x); the λ are elements of theCartan sub-algebra, which is the set of mutually commuting generators in the group.We introduce an effective potential for the phases asZ(˜λ) =ZAµ(x,β)=+Aµ(x,0) DAµ(x, τ) e−S δ ˜λ −Z d3xVλ(x)!.

(3)DAµ is the functional measure for the gauge fields, including gauge fixing and ghosts,S is the gauge field action, and V is the volume of space. We have chosen to use a“constraint” effective potential [11]; a more standard form, using an external sourceand then Legendre transformation, could also be used.

From Z(˜λ) the free energy, asa function of the ˜λ, is F(˜λ) = −log(Z(˜λ)).When the phases vanish, F(0) reduces to the usual free energy at a temperatureT, in the absence of any background field. (This is seen most easily from (14) below.

)It is for this reason that we constrain the phases of the eigenvalues instead of theeigenvalues themselves.Since the ˜λ’s are phases, the free energy is periodic: for˜λij = ˜λiδij, F(˜λ) is unchanged when any single element is shifted by a multiple of 2π,F(˜λi + 2π) = F(˜λi).In a pure gauge theory there is a further degeneracy in F(˜λ). The gauge poten-tials remain periodic under certain aperiodic gauge transformations, as long as theaperiodicity is a constant element in the center of the group [1].

For an SU(N) gauge2

group the center is Z(N) , and the aperiodic gauge transformations satisfy Ω(x, β) =eiθℓΩ(x, 0), where θℓ= 2πℓ/N, and ℓis an integer = 0, 1, ...(N −1). Under such gaugetransformations every phase transforms by the same constant, λi(x) →λi(x) + θℓfor all i = 1...N. This implies that the effective potential automatically posseses aZ(N) symmetry, F(˜λ + θℓ) = F(˜λ) (here θℓdenotes a diagonal matrix where eachelement = θℓ; this notation is used later, and should be clear from the context).Before turning the effective potential into a partition function, it is worth review-ing the results of perturbative calculations [2-7].

To compute one expands about aconstant, background field A0 = Acl0 + Aqu0 , where Acl0 = ˜λ/g. A convenient gauge iscovariant background field gauge, where the gauge fixing term in the lagrangian isLcovariant = 1ξ trDclµ Aquµ2,(4)Dclµ = ∂µ −ig[Aclµ, ].

At one loop order the free energy is independent of the gaugefixing parameter ξ, Fone loop(˜λ) = 2 tr log(−(Dclµ )2); this is the free energy for twospatially transverse gluons in the background field Acl0 . The minimum is at ˜λ = 0,Fone loop(0) ≤Fone loop(˜λ), plus Z(N) transforms thereof.At two loop order, as a function of Acl0 the free energy is gauge dependent.

Belyaev[6,7] noted that this happens because at one loop order fluctuations in the gauge field— in particular the static components of Aqu0 which do not commute with Acl0 — alterthe relationship between ˜λ and Acl0 from that at tree level. This feeds back into thefree energy at two loop order.

As a function of the ˜λ, the free energy is independentof ξ, with a minimum for ˜λ = 0, Ftwo loop(0) ≤Ftwo loop(˜λ).There are “unitary” gauges in which the relationship Acl0 = ˜λ/g is unchanged toany loop order. A reasonable guess is static gauge [12], imposing ∂0A0 = 0.

This is notadequate, however, for static gauge has a residual gauge freedom of performing staticgauge transformations. Dividing up A0 = Acl0 +Aqu0 , some static gauge transformationsrotate the background field Acl0 , which is inconsistent.

A gauge which avoids this isstatic background gauge, obtained by adding the termLstatic = 1ξ tr 1ξ′Dcl0 Aqu0 + ∂iAqui!2(5)to the lagrangian.Letting the gauge parameter ξ′ →0 fixes Dcl0 Aqu0= 0.Thisrestricts all fluctuations in Aqu0to be both static and in directions which commutewith Acl0 . The value of the other gauge fixing parameter ξ is arbitrary; ξ removes the3

remaining gauge degeneracy under static gauge transformations which commute withAcl0 .The calculation of the free energy to one loop order in static background gauge isilluminating. At leading order the ghost determinant depends on the background fieldas det(−(Dcl0 )2).

In static background gauge the square root of the ghost determinantequals a product of Vandermonde determinants for the Wilson line,det(Dcl0 ) =Yx|V(˜λ)|2,V(˜λ) =Yi

Even in other regularizationschemes, however, the product of Vandermonde determinants from the static back-ground ghosts do not contribute to the free energy F(˜λ). One factor cancels againstthe contribution of the spatially longitudinal mode of the gluon.1 The second fac-tor cancels against terms from the A0 components of the propagator, which give afinite but nonzero contribution in the limit ξ′ →0.

After these cancellations the twospatially transverse modes give the same result as found in covariant gauge.To turn the path integral of (3) into a partition function we follow Rossi and Testa[8]. First we transform to A0 = 0 gauge.

The gauge transformation which accom-plishes this transformation is the “partial” Wilson line, Ω(x, τ) = Pexp(igR τ0 A0(x, τ ′)dτ ′).Many factors can be ignored in going to A0 = 0 gauge. The Fadeev-Popov deter-minant is det(∂0), which is independent of temperature and the background field.Similarly, when 0 ≤τ < β, all terms in the measure from the gauge transformationΩ(x, τ) can all dropped.

The sole exception is the gauge transformation for the lastslice of euclidean time, Ω(x, β). This must be retained because it affects the boundaryconditions: the boundary conditions are aperiodic, Ai(x, β) = +AΩ(x,β)i(x, 0), while1 This cancellation is similar to that found by Weiss [2], who imposed the condition that A0 isstatic and diagonal by fiat.

There is no ghost per se, but the measure of the functional integralincludes a single factor of the Vandermonde determinant squared at each point in space. He obtainsthe correct free energy at one loop order, where the product of Vandermonde determinants inthe measure cancels against the contribution of the spatially longitudinal gluons.

One can check,however, that his method does not give the correct form of Gauss’ law in the Cartan sub-algebra.The problem is that the gauge condition ∂0A0 = 0 can only be imposed after allowing arbitraryvariations in A0, not before. Imposing static background gauge as in the text, by letting ξ′ →0,avoids this difficulty.4

the Wilson line L(x) →Ω(x, β). Relabeling Ω(x, β) as Ω(x), in A0 = 0 gauge thepath integral isZ(˜λ) =ZDΩ(x)ZAi(x,β)=+AΩi (x,0) DAi(x, τ) e−S(A0=0) δ ˜λ −Z d3xVλ(x)!.

(7)As before we decompose Ω(x) = U(x)Λ(x)U†(x), Λ(x) = eiλ(x). In terms of thesevariables the measure DΩ= DU Dλ |V(λ)|2, where the measure naturally includesthe square of the Vandermonde determinant [13].

Thus in A0 = 0 gauge the constrainton the Wilson line becomes a constraint on the final gauge transformation, Ω(x).An advantage of A0 = 0 gauge is that the canonical structure is manifest [8,9].Thus the functional integral in (7), written in imaginary time, is equal to the partitionfunction,Z(˜λ) =ZDΩ(x)XAi(x)⟨Ai(x)| e−βH |AΩi (x)⟩δ ˜λ −Z d3xVλ(x)!. (8)All fields Ai(x, t) are functions of a fixed time t; this time is arbitrary, and so thedependence upon it is suppressed.

We assume canonical commutation relations be-tween Ai(x) and the electric field Ej(x), [Aai (x), Ebj(y)] = i δab δij δ3(x −y). H is thetotal Hamiltonian, with the sum over all eigenstates |Ai(x)⟩; |AΩi (x)⟩is the gaugetransformation of a state under Ω(x).To obtain a more useful form of the partition function we introduce the generatorsof gauge transformations.

Given Ω(x), let Ω= eiω, and define the operatorG(ω) = −2gZd3x tr ((Diω(x)) Ei(x)) . (9)The canonical commutation relations imply that these generators form a representa-tion of the Lie algebra,[G (ω1) , G (ω2)] = G ([ω1, ω2]) .

(10)By the operator identity AΩi (x) = e−iG(ω)Ai(x)eiG(ω), the gauge transformed state isgiven by|AΩi (x)⟩= eiG(ω)|Ai(x)⟩. (11)Since the operators G(ω) generate gauge transformations, they commute with theHamiltonian, [G(ω), H] = 0.5

We write the eigenvalues λ(x) = ˜λ + λq(x), so that the constraint becomesR d3x λq(x) = 0. This constraint implies that there is no constant mode in λq(x),and is automatically satisfied by requiring that λq(x) vanish at spatial infinity.

Inthis way we trade the constraint for a boundary condition,Z(˜λ) =Zλq(∞)=0 DU(x) Dλq(x) |V(˜λ+λq(x))|2XAi(x)⟨AUi (x)| e−βH+iG(˜λ)+iG(λq) |AUi (x)⟩,(12)To obtain this we start with |AΩ(x)⟩, and by using Ω= UΛU†, undo the last twogauge transformations. U† is carried through to change ⟨Ai(x)| into ⟨AUi (x)|.

Thegenerators for the gauge transformations in the Cartan sub-algebra are left as is. Thefactor of G(λq(x)) generates a local, that for G(˜λ) a global, gauge transformation inthe Cartan sub-algebra.

The generator of the global gauge transformation can bewritten as a chemical potential for global color charge, Q:G(˜λ) = 2 tr(˜λ Q),Q = −i gZd3x [Ai(x), Ei(x)] . (13)The appearance of the background field as an imaginary chemical potential is notsurprising: this is obvious at tree level, using the canonical formalism to expandabout a constant field Acl0 = ˜λ/g; see, e.g., ref.’s [3] and [4].

What is not obvious,and which (12) and (14) demonstrate, is how to write the partition function Z(˜λ) ina gauge invariant manner beyond tree level.It is worth checking that to leading order the partition function in (12) gives thesame free energy as found in the imaginary time formalism. The integrations over thegauge transformations U(x) and λq(x) can be ignored to this order.

Thus the onlychange from calculating the free energy in zero field is the presence of an imaginarychemical potential for global color charge. Partition functions of this form have beenstudied [10]; from these calculations we see that the spatially transverse gluons givethe same free energy as found to one loop order in the imaginary time formalism.

Thespatially longitudinal mode, with zero energy, cancels the product of Vandermondedeterminants from the measure,Qx |V(˜λ + λq(x))|2 ≈Qx |V(˜λ)|2.Since λq vanishes at spatial infinity we can integrate G(λq) by parts to obtainG(λq) = 2R d3x tr(λq DiEi)/g . This presumes that the gauge field itself satisfies theappropriate boundary conditions: Ai(x) ∼1/r and Ei(x) ∼1/r2 as x →∞suffice.The only other place where λq enters is through the infinite product of Vandermondedeterminants, Qx |V(˜λ+ λ(x))|2.

As remarked previously, however, this infinite prod-uct vanishes if we adopt dimensional regularization, which we do. Then integration6

over λq is trivial, generating a projection operator, PC = Qx δ(tr(taCDiEi(x))) , wherethe taC are the N −1 elements of the Cartan sub-algebra. This imposes Gauss’ law inthe Cartan sub-algebra at each point in space.

WhenceZ(˜λ) =ZDU(x)XAi(x)⟨AUi (x)| e−βH + 2 i tr(˜λQ) PC |AUi (x)⟩. (14)This form of the partition function involves only states with positive norm, the|AU(x)i(x)⟩, and a positive measure of integration over U(x).

The background fieldenters only through the global color charge, e2 i tr(˜λQ). While this factor is imaginary,because the sum always includes states with equal and opposite charge, the total isnecessarily real, so e2 i tr(˜λQ) ∼cos(2 tr(˜λQ)).

Since the cosine function is less than orequal to one, a nonzero field ˜λ decreases the partition function, Z(˜λ) ≤Z(0), and soincreases the free energy,F(0) ≤F(˜λ) ,(15)which concludes our proof.Our argument is inspired by that of Vafa and Witten [14] for the θ-angle in QCD.Nonzero θ appears in the euclidean path integral as a phase factor, eiθQtop, whereQtop is the topological charge.If the functional measure is positive definite, thisimplies that θ = 0 and π minimize the corresponding free energy. In our case weneed positivity for the sum over states in the partition function (which is why wetake A0 = 0 gauge) as the background field appears as a phase factor for global colorcharge.A delicate point is the use of dimensional regularization to eliminate the infiniteproduct of Vandermonde determinants.

Based upon calculations to one loop order,where the product of Vandermonde determinants cancels in F(˜λ), we suggest thatthis cancellation persists to arbitrary loop order. There is a general reason why itshould.

Consider Z(˜λ) in (1), and then integrate over all ˜λ; the delta function dropsout to give the free energy in zero field. (Integrating (14) with respect to ˜λ givesa constraint of zero color charge; since in infinite volume the minimum is alwaysat ˜λ = 0, this constraint is inconsequential.) But the appropriate measure for theintegration over ˜λ, D˜λ |V(˜λ)|2, already includes a single factor of the square of theVandermonde determinant.

Thus there should not be any additional factors in Z(˜λ),infinite or not.It is trivial to extend the above to include the effects of matter fields. Fermionfields, for instance, are added to the euclidean functional integral, (3), through an-7

tiperiodic boundary conditions.ψ(x, β) = −ψ(x, 0).The above derivations gothrough with minor modifications: the states include both gauge fields and fermions,|Ai(x), ψ(x)⟩, and the global color charge includes the contribution of the matterfields as well. But again we conclude that F(0) ≤F(˜λ).The partition function also provides insight into the global Z(N) symmetry.

If allmatter fields lie in the adjoint representation, the free energy, and so the vacua, areZ(N) symmetric, F(0) = F(θℓ) for all ℓ= 0, 1, ...(N −1). This Z(N) symmetry ismanifest from the form of the partition function in (14).

The factor of e2 i tr(θℓQ) =eiG(θℓ) generates a global Z(N) gauge transformation on the states by Ω= eiθℓ. Asan element of the center of the group this transformation leaves gluon (and otheradjoint) states unaltered, e2 i tr(θℓQ)|AUi ⟩= |AUi ⟩.If there are matter fields in the fundamental representation, the Z(N) symmetryis lifted at one loop order: for ℓ̸= 0, F(0) < F(θℓ), as the trivial vacuum, ˜λ = 0, isthe unique vacuum (modulo the usual factors of 2π because ˜λ is a phase).

In termsof the partition function, the Z(N) degeneracy is lifted because while states in theadjoint representation don’t change under Z(N) global gauge transformations, statesin the fundamental representation do:e2 i tr(θℓQ)|AUi , ψU⟩= |AUi , eiθℓψU⟩̸= |AUi , ψU⟩. (16)In this way, we see that while the Wilson line is defined originally in euclidean space-time, its transformation under (aperiodic) global Z(N) gauge rotations directly re-flects the transformations of states, in the partition function, under global Z(N) gaugerotations.We conclude by discussing the nature of states which contribute to the partitionfunction in (14).

Since by definition Z(˜λ) is gauge invariant, the sum in states in (14)must be as well. A (static) gauge transformation V (x) changes Ai →AVi ; both thebra and ket states change, ⟨AUi | →⟨AV Ui| and |AUi ⟩→|AV Ui⟩.

As the Haar measure isinvariant under left multiplication, though, by U(x) →V †(x)U(x) we can eliminateany such V (x).While the sum over states is gauge invariant, the states which contribute are not.This is very different from the computation of the free energy in zero field, F(0).Begin with (4), and just drop the constraint altogether. Then F(0) involves states ofthe form|Ai(x)⟩inv =ZDΩ(x) |AΩi (x)⟩.

(17)8

By the properties of the Haar measure, these states are manifestly gauge invariant[1,8], |AVi ⟩inv = |Ai⟩inv. In nonzero field, however, the presence of the constraint onthe eigenvalues implies that more than just gauge invariant states contribute to thesum.Alternately, consider how Gauss’ law works.

In zero field it is easy to start from(8) and show that the integral over Ωimposes Gauss’ law in all directions of thegroup [1,8,9]. This can be seen, albeit less directly, from (14): for ˜λ = 0, take eachstate, say the bra ⟨AUi |, and undo the gauge transformation U(x) by writing it as⟨Ai|e−iG(u), U = eiu.

Doing this for both bra and kets, the projector for the Cartansub-algebra becomes e−iG(u)PCeiG(u). This is the gauge transformation of Gauss’ lawin the Cartan sub-algebra, so integration over the U(x) imposes Gauss’ law in alldirections of the group algebra.These manipulations fail in nonzero field, ˜λ ̸= 0.

The problem is that the generatorfor the local gauge transformation, G(u), need not commute with that for the globalgauge transformation, G(˜λ): there are always elements u(x) for which [u(x), ˜λ] ̸= 0,so [G(u), G(˜λ)] ̸= 0. Thus in nonzero field, only the components of Gauss’ law in theCartan sub-algebra are set to zero, and not the remaining components.This feature is generic to any effective potential which is a function of the Wilsonline.

Consider, for example, the potential for the real part of the trace of the Wilsonline. In euclidean spacetime,Ztr(J) =ZAµ(x,β)=+Aµ(x,0) DAµ(x, τ) e−S−J Rd3x Re(tr L),(18)using a source J coupled to the trace of the Wilson line.

Using the above techniquesthis equals the partition functionZtr(J) =ZDΩ(x)XAi⟨Ai(x)| e−βH−J Rd3x Re(tr Ω) |AΩi (x)⟩. (19)Again, while the sum over states is gauge invariant, the individual states are not: thesource dependent state|Ai(x), J⟩=ZDΩ(x) e−J Rd3x Re(tr Ω)|AΩi (x)⟩(20)is not gauge invariant: |AVi , J⟩̸= |Ai, J⟩when J ̸= 0.This behavior is not entirely unexpected.

After all, the vacuum expectation valueof the trace of the Wilson line, ⟨tr L(x)⟩= ∂ln(Z(J))/∂J(x)|J=0, is the wave function9

for an infinitely heavy test quark [1]. In terms of Gauss’ law, a single Wilson linecorresponds to the insertion of a point source of unit strength at that point [1].

Asource for the (trace of the) Wilson line, spread out over all of space, then modifiesGauss’ law everywhere.From (19), when J(x) ̸= 0 no component of Gauss’ lawvanishes, not even those in the Cartan sub-algebra. Amusingly, the right hand side ofGauss’ law, which is proportional to the source J, is only real if J is purely imaginary.Partition functions similar to that in (14) have been studied before [10].

Thedifference is that these works all assume that only gauge invariant states, as in (17),contribute:Zinv(˜λ) =XAi(x)⟨Ai(x)|inv e−βH+2 i tr(˜λQ)|Ai(x)⟩inv ,(21)At zero field everything reduces to the standard free energy: Z(0) = Zinv(0). SinceZinv contains no term in the measure for the Vandermonde determinant, at one looporder Z(˜λ) and Zinv(˜λ) differ by a product of Vandermonde determinants, as thespatially longitudinal gluons give 1/ Qx |V(˜λ)|2.

This leads one to suspect that per-haps to arbitrary loop order, Z(˜λ) and Zinv(˜λ) differ only by terms which vanish indimensional regularization.Nevertheless, (21) has one serious drawback: in euclidean spactime it does not cor-respond to the functional integral for a gauge invariant potential. Using the definitionof the invariant state |Ai(x)⟩inv,Zinv(˜λ) =ZDΩ1(x)ZDΩ2(x)XAi(x)⟨Ai(x)| e−βH|AΩfi (x)⟩,Ωf(x) = Ω1(x)ei˜λΩ†2(x) .

(22)This is similar to the partition function in (8), with the replacement of the gaugetransformation Ωby Ωf. In (8) the background field ˜λ is related to a gauge invariantquantity, the spatial average over the phases of the eigenvalues of Ω(x).In (22),however, ˜λ is not related to anything gauge invariant, since the eigenvalues of Ωf =Ω1ei˜λΩ†2 are not ei˜λ.

Computing the euclidean path integral which corresponds to(22) [1] leads to an expansion about constant Acl0 = ˜λ/g, which is gauge dependentbeyond leading order.Consider the consequences if, by dint of prejudice, one insists that only gaugeinvariant states should contribute to the partition functions of gauge theories. Thenconstruct the effective potential for the trace of the Wilson line, as in (19), everywherereplacing the states |Ai⟩with gauge invariant states, |Ai⟩inv.

But as |AΩi ⟩inv = |Ai⟩inv,this partition function gives a trivial potential for the trace of the Wilson line,10

Zinv(J) =QxR DΩ(x)e−J Rd3x Re(tr Ω): this potential is independent of temperatureor the field content, and as an infinite product, vanishes in dimensional regulariza-tion. This is untenable, manifestly in contradiction with numerical simulations ofeuclidean lattice gauge theories.Partition functions as in (14) do produce novel thermodynamic behavior.Inordinary partition functions each state makes a positive contribution to Z, whichimplies that the pressure is positive.

In nonzero field, however, as the sum over statesincludes a factor of cos(2 tr(˜λQ)), states with nonzero charge can make a negativecontribution to Z(˜λ), and consequently the pressure can be negative. For instance,even for a single scalar field coupled to an abelian gauge field, there are regions of ˜λin which the pressure is negative.Dixit and Ogilvie [3] observed that in theories with dynamical fermions, whileZ(N) transforms of the usual vacuum have greater free energy than the trivial vac-uum, they are often metastable.

The cosmological implications of such Z(N) “bubbles”were developed by Ignatius, Kajantie, and Rummukainen [2]. In ref.

[4], however,it was pointed out that the pressure of the metastable Z(N) state can be negative,depending upon matter content of the theory. This includes the physically interest-ing case of three colors and six massless flavors, as applies at temperatures above therestoration of the SU(2) × U(1) symmetry.

While the pressure of W, Z, and Higgsbosons acts to make the total pressure positive, it is rather disquieting to find thatthe pressure of a subsystem is negative. On these, and other grounds, the authors of[4] argue that the metastable Z(N) states are not thermodynamically accessible.We suggest the contrary.

Surely a system with negative pressure is not thermo-dynamically stable; indeed, it will not even be metastable for long. But we proposethat if a path integral is gauge invariant and well defined in euclidean spacetime, thenthe corresponding partition function, as in (14) and (19), do represent physically re-alizable systems.

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