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Casimir effect of strongly interacting scalar fields1

arXiv:hep-ph/9307258v1 13 Jul 1993UNIT¨U-THEP-8/1993July 12, 1993Casimir effect of strongly interacting scalar fields1K. Langfeld, F. Schm¨user, H. ReinhardtInstitut f¨ur theoretische Physik, Universit¨at T¨ubingenD–72076 T¨ubingen, GermanyAbstractNon-trivial φ4-theory is studied in a renormalisation group invariant ap-proach inside a box consisting of rectangular plates and where the scalarmodes satisfy periodic boundary conditions at the plates.

It is found that theCasimir energy exponentially approaches the infinite volume limit, the decayrate given by the scalar condensate. It therefore essentially differs from thepower law of a free theory.

This might provide experimental access to prop-erties of the non-trivial vacuum. At small interplate distances the system canno longer tolerate a scalar condensate, and a first order phase transition tothe perturbative phase occurs.

The dependence of the vacuum energy densityand the scalar condensate on the box dimensions are presented.1 Supported by DFG under contract Re 856/1 −11

1IntroductionThe Casimir effect [1, 2, 3, 4] in quantum field theory is the change of the vac-uum energy density due to constraints on the quantum field induced by boundaryconditions in space-time. The contribution to the energy density by the quantumfluctuation of the electromagnetic field was experimentally observed by Sparnaay [5]in 1958, thus verifying its quantum nature.

Following this observation the Casimireffect was extensively studied, the renormalisation procedure that must be used inorder to extract physical numbers out of divergent mode sums, being of particularinterest. This procedure is most elegantly formulated in a path-integral approach [6],and leads to a full understanding of the Casimir effect for non-interacting quantumfields.

Perturbative corrections to the free Casimir arising from a weak interactionof the fluctuating fields can also be obtained [7]. It was shown that the net effectof the boundaries is to produce a topolocigal mass for the fluctuating modes [8].

Inthe recent past there has been a renaissance of the Casimir effect due to its broadspan of applications, which range from gravity models [9] to QCD bag models [10] tonon-linear meson-theories describing baryons as solitons [11]. A closely related sub-ject is Quantum Field Theory at finite temperature since it can be described in thepath-integral formalism by implementing periodic boundary conditions in Euclideantime direction [12].

Despite these many different applications, it is possible to un-derstand the basic features of the Casimir effect by investigating a scalar theory. Itis also of general interest to study φ4-theory due to its important applications, e.g.in the Weinberg-Salam model of weak interactions (see e.g.

[13]) and in solid statephysics [14]. Many different approaches [15, 16] to strongly interacting φ4-theorywere designed to understand its non-trivial vacuum structure.Even though the Casimir effect of free quantum fields is well understood, thereis not yet an understanding of the Casimir effect for strongly interacting fields.This is simply due to the lack of knowledge of the true vacuum of an interactingquantum theory.

Recently, a non-perturbative path integral approach to φ4-theoryhas yielded some insight into the vacuum structure of the strongly interacting scalartheory [17, 18]. In particular, it was found that its perturbative phase is unstable(at zero temperature) because a second phase with non-vanishing scalar condensatehas lower vacuum energy density [17].

In this phase, the connection between thescalar condensate and the vacuum energy density, which is provided by the scaleanomaly, has been verified by an explicit calculation [18]. The structure of thisnew phase, describing strongly interacting scalar modes, was also investigated atfinite temperature [18].It was found that at a critical temperature the energydensities of the non-trivial and perturbative phases are equal, and the non-trivialphase undergoes a first order phase transition to the perturbative one.Using these results it is possible to study the Casimir effect of strongly interactingscalar fields.

Since the non-trivial phase provides an intrinsic energy scale (i.e. the2

magnitude of the scalar condensate at zero temperature), one expects deviationsof the Casimir force from the free field law. This presumably provides access tonon-perturbative vacuum properties.In this paper, we investigate the non-trivial phase of four dimensional φ4-theory in arectangular box consisting of p (< 4) pairs of oppositely layered plates separated bya distance ap, with the scalar modes satisfying periodic boundary conditions at theplates.

We shall find at large interplate distances, that the Casimir energy decaysexponentially with increasing distance, the decay rate given by the magnitude of thescalar condensate. At small distances, the field theory no longer tolerates a scalarcondensate, and the perturbative phase is adopted.The paper is organised as follows: in the second section we briefly review the Casimireffect of a free theory and the recently proposed non-perturbative approach [17, 18]to φ4-theory.

The renormalisation procedure is discussed and renormalisation groupinvariance is shown. In the subsequent section results are presented.

The Casimir en-ergy as a function of large (compared with the scalar condensate) interplate distanceis obtained analytically, and the deviations from the energy in a free field theoryare discussed. The phase-transition from the non-trivial vacuum to the perturbativephase at small interplate distances is studied and the vacuum energy density andthe scalar condensate is calculated as function of the plate distances.

Discussionsand concluding remarks are given in the final section.2The Casimir effect of scalar fieldsφ4-theory is described by the Euclidean generating functional for Green’s functions,i.e.Z[j] =ZDφ exp{−Zd4x (12∂µφ∂µφ + m22 φ2 + λ24φ4 −j(x)φ2(x) ) } ,(1)where m denotes the bare mass of the scalar field and λ the bare coupling strengthof the φ4-interaction. j(x) is an external source for φ2(x) which is introduced sothat we can derive the effective potential [19, 20] of the composite field φ2 later on.It was observed in [18] that it is more convenient to use the effective potential ofφ2 to study the phase structure of the theory.

In particular, its minimum value isthe vacuum energy density and thus provides access to the Casimir effect, if it iscalculated by imposing adequate boundary conditions to the scalar modes. For theseinitial investigations we adopt the simplest geometry and consider a rectangular boxconsisting of p (< 4) pairs of oppositely layered plates separated by distances ap.We expect that the Casimir energy will not be sensitive to the detailed shape of thefinite volume as is known in a free theory [21].

The integration over the field φ in (1),3

only extends over configurations which satisfy periodic boundary conditions at theplates. In the case of Dirichlet or Neumann boundary conditions, surface counterterms must be added to (1).

In that case the results would sensitively depend onthe physical structure of the surface, and such effects are beyond the scope of thispaper.The effective action is defined by a Legendre transformation of the generating func-tional Z[j], i. e.Γ[φ2c] := −ln Z[j] +Zd4x φ2c(x)j(x) ,φ2c(x) := δ ln Z[j]δj(x). (2)From here the effective potential U(φ2c) is obtained by restricting φc to constantclassical fields (Γ[φ2c = const.] =R d4x U(φ2c)), which are obtained for a constantexternal source j.

The minimum value of the effective potential Umin is the vacuumenergy density and is obtained from (2) at zero external source, i.e.dUdφ2c|φ2c=φ2c 0 = j = 0 . (3)The minimum classical configuration φ2c 0 represents the scalar condensate.2.1Equivalence of effective action and sum of zero-pointenergiesIn this subsection we review the Casimir effect of a free scalar theory (λ = 0) usingSchwinger’s proper-time regularisation.

We demonstrate that the minimum of theeffective potential U coincides with the mode sum usually considered when studyingthe Casimir effect [1, 2, 3, 4]. This equivalence was also obtained by using anotherregularisation scheme [2], and previously observed with proper-time regularisationin the context of chiral solitons [22].The minimum of the effective potential of a free scalar theory isUmin =12TV3Tr ln(−∂2 + m2) ,(4)where T is the Euclidean time interval, and V3 is the space volume.

The trace in(4), extending over all modes satisfying periodic boundary conditions, is a divergentobject and needs regularisation. For definiteness we use Schwinger’s proper-timeregularisation, but note, however, that the specific choice of the regularisation pre-scription has no influence on the renormalised (finite) result (e.g., compare [20] and[23]).

In proper-time regularisation, the vacuum energy density becomesUmin = −12TV3TrZ ∞1/Λ2dss e−s(−∂2+m2) ,(5)4

where Λ is the ultraviolet cutoff.The trace over the temporal degree of freedom can be easily performed, i.e.,Umin = −12V3Z dk02π TrVZ ∞1/Λ2dss e−s(k20+E) ,(6)The trace TrV extends over the spatial degrees of freedom and E is the energyobtained from the eigenvalue equation(−∇2 + m2)φ(x) = E2φ(x) ,(7)where the eigenfunction φ satisfies periodic boundary conditions. If the k0-integra-tion in (6) is performed, a partial integration in the s-integral yieldsV3Umin =12√πTrV {E Γ(12, E2Λ2 )} −Λ2√π TrV exp{−E2Λ2 } ,(8)where Γ( 12, x) is the incomplete Γ-function.

The first term of (8) is precisely the modesum 12TrV E in cutoffregularisation, with the particular cutofffunction1√πΓ( 12, E2Λ2 )provided by Schwinger’s proper-time regularisation. In the limit of large Λ, the sec-ond term only contributes a constant to the action which is subtracted by demandingthat the Casimir energy approaches zero for large interplate distances.In order to illustrate the equivalence of the mode sum approach and the approachprovided by the effective potential, we calculate the Casimir energy for a masslessscalar particle in a box consisting of p pairs of rectangular plates and in d space-timedimensions.

In this case we have−ln Z = −Ld−p2Zdd−pk(2π)d−p∞X{ni}=−∞Z ∞1/Λ2dss exp{−s[k2 +pXi=1n2i (2πai)2]} ,(9)where ai, i = 1 . .

. p is the distance of the hyperplanes in ith direction and L ≫aiis the length of the box of the unconstrained modes.The integration over thecontinuous degrees of freedom can be performed in a straightforward manner, i.e.,−ln Z = −Ld−p21(2√π)d−p∞X{ni}=−∞Z ∞1/Λ2dss1+ d−p2e−sPn2i ( 2πai )2 ,(10)In order to extract the ultra-violet divergences, we apply Poisson’s formula∞Xn=−∞f(n) =∞Xν=−∞c(ν) ,withc(ν) =Z ∞−∞dn f(n) ei2πνn .

(11)Rewriting∞Xn=−∞e−sn2( 2πa )2 =a2√πs∞Xν=−∞e−ν2 a24s(12)5

equation (10) becomes−ln Z = −Ld−p21(2√π)d−pZ ∞1/Λ2dss1+ d−p2pYi=1[ai2√πs∞Xνi=−∞e−ν2ia2i4s ] . (13)Note that the ultra-violet behaviour is dominated by the integrand at small s andthe only divergences come from the term with all νi are zero.

The divergent term−ln Zdiv = −Ld−p21(2√π)d−pZ ∞1/Λ2dss1+ d−p2pYi=1[ai2√πs] ,(14)is proportional to the d-dimensional volume V and a pure constant, which can beabsorbed by a redefinition of the action. After the substitution s →1/s, the s-integral can be performed in (13), yielding for the finite part in the limit Λ →∞−1V ln Z = Umin = −121πd/2 Γ(d2) Z(a1 .

. .

ap, d) ,(15)whereZ(a1 . .

. ap, d) =∞X{νi}=−∞′1(a21ν21 + .

. .

+ a2pν2p)d2 ,(16)is the Epstein Zeta-function (the prime indicates that the contribution with all νi = 0is excluded from the sum). For p = 1 and four space-time dimensions (d = 4) oneobtains the analytic result for the vacuum energy density and the Casimir energyEc, respectively, i.e.,Umin = −1π21a4Γ(2)ζ(4) ,Ec = V3Umin = −π2L290a3 ,(17)where ζ(s) =P∞1 n−s is Riemann’s ζ-function.

This is precisely the result usuallyobtained by evaluating the mode sum of zero point energies [2].2.2Non-trivial φ4-theory with boundary conditionsIn this subsection we describe the non-perturbative approach to φ4-theory providedby the modified loop expansion [17], taking into account the constraints on thescalar field imposed by boundary conditions. We demonstrate that the renormali-sation procedure is not affected by the presence of a rectangular box, implying thatrenormalisation group invariance is preserved as it is in the infinite volume limit(ai →∞).6

The modified loop expansion [17] is based on a linearisation of the φ4-interaction inthe path-integral (1) by means of an auxiliary field χ(x)Z[j] =ZDφ Dχexp{−Zd4x [ 12∂µφ∂µφ +(18)6λχ2(x) + [m22 −iχ(x)]φ2(x) −j(x)φ2(x)] } .This linearisation was first proposed in [24]. The integral over the fundamental fieldφ is then easily performed, yieldingZ[j]=ZDχ exp{−S[χ, j]} ,(19)S[χ, j]=6λZd4x χ2 + 12Tr(R) ln D−1[χ, j] ,(20)D−1[χ, j]xy=(−∂2 + m2 −2iχ(x) −2j(x))δxy .

(21)The trace Tr(R) extends over all eigenmodes of the operator D−1[χ, j] which satisfythe periodic boundary conditions and the subscript (R) indicates that a regulari-sation prescription is required. Note that the boundary conditions to the field φdo not give rise to any constraint for the auxiliary field χ.

The approach of [17] isdefined by a modified expansion with respect to the field χ around its mean fieldvalue χ0 defined byδS[χ, j]δχ(x) |χ=χ0 = 0 . (22)The modified loop expansion of [17, 18] coincides with an 1/N-expansion of O(N)symmetric φ4-theory [25] for N = 1, implying that the convergence of the expansionis doubtful.

However, it was seen in zero dimensions that the effective potential ofthis approximation rapidly converges to the exact one obtained numerically. Recentresults show that the same is true for four dimensional φ4-theory [26].

Reasonableresults are obtained even at mean-field level. At this level we obtain from (18)−ln Z[j](a1 .

. .

ap)=(23)Zd4x {−32λ(M −m2 + 2j)2} + 12Tr(R) ln(−∂2 + M) ,where M is related to the mean field value χ0 by χ0 = i(M −m2 +2j)/2. The meanfield equation for χ0 (22) can be recast into an equation for M, i.e.,δ ln Z[j]δM= 0 .

(24)For a constant external source j, this equation is satisfied for constant M.7

The only effect of the rectangular box is contained in the loop contribution, whichis, in Schwinger’s proper-time regularisation (d = 4)L=12Tr(R) ln(−∂2 + M)(25)=−L4−p2Zd4−pk(2π)4−pX{ni}Z ∞1/Λ2dssexp{−s[4−pXl=1k2l + M +Xi(2πai)2n2i ]} ,with i = 1 . .

. p and L the linear extension in the unconstrained directions.

Thesum over the unconstrained modes (k-integral) can be easily performed. ApplyingPoisson’s formula (11) to extract the divergent terms as we did for the free theory(section 1) we obtainL = −V32π2M2 Γ(−2, MΛ2) −V32π2X{νi}′ Z ∞1/Λ2dss3 e−sM exp(−Xia2i4sν2i ) ,(26)where Γ is the incomplete Γ-function, V the space-time volume and the prime in-dicates that the contribution with all νi = 0 is excluded.

This implies that thesecond term of the right hand side of (26) is ultraviolet finite, so we can remove theregulator in this term (Λ →∞). Using the asymptotic expression of the incompleteΓ-function, we findL =V32π2{MΛ2 + 12M2(ln MΛ2 −32 + γ)} −V32π2 F3(M, a1 .

. .

ap) ,(27)where γ = 0.577... is Euler’s constant and the function F3 is defined byFǫ(M, a1 . .

. ap) =X{νi}′ Z ∞0dssǫ e−sM e−Pia2i4s ν2i(28)with ǫ = 3.

The first term on the right hand side of (27) is precisely the effectivepotential in the infinite volume limit since the function F3 vanishes for ai →∞.The second term in (27) is thus the modification of the effective potential due to thepresence of the plates. Note that this term is finite, implying that the boundariesdo not affect the renormalisation procedure.

This is the desired result.Following the renormalisation scheme given in [18], we absorb the divergences in thebare parameters λ, m, j by setting6λ +116π2 (ln Λ2µ2 −γ + 1)=6λR(29)6λj −3m2λ−132π2Λ2=6λRjR −3m2RλR(30)j −m2=0 ,(31)8

where µ is an arbitrary renormalisation point and a subscript R refers to the renor-malised quantities. Later we will check that physical quantities do not depend onµ.

In the following, we consider the massless case mR = 0. The coupling strengthrenormalisation in (29) was earlier used by Coleman et al.

[24] and, as pointed outby Stevenson, it implies that the bare coupling becomes (infinitesimally) negative,if the regulator Λ is taken to infinity [16]. It was shown that this behaviour of thebare coupling strength is hidden in the standard perturbation theory [17].

In fact,we have from (29)λ =λR1 −β0λR(ln Λ2µ2 −γ + 1) ,β0 =196π2(32)implying λ →0−for Λ →∞, whereas in contrast an expansion of (32) with respectto the renormalised coupling strength, i.e.λ = λR(µ) [1 + β0 λR(ln Λ2µ2 −γ + 1) + O(λ2R)](33)suggests that λ →+∞, if Λ →∞.Inserting (29-31) and L from (27) in (23) one obtains−1V ln Z[j](a1 . .

. ap)=−32λRM2 −6λRMjR + α2 M2(ln Mµ2 −12)(34)−α F3(M, a1 .

. .

ap)where α = 1/32π2, and M is defined by the mean field equation (24), i.e.−3λRM −6λRjR + αM(ln Mµ2 −12) + α F2(M, a1 . .

. ap) = 0 .

(35)It is now straightforward to perform the Legendre transformation (2). The finalresult for the effective potential isU(φ2c) = α2 M2(ln Mµ2 −12) −32λRM2 −α F3(M, a1 .

. .

ap) ,(36)whereφ2c =1Vδ ln Z[jR]δjR=6λRM . (37)The effective potential is renormalisation group invariant, since a change in therenormalisation point µ can be absorbed by a change of the renormalised couplingstrength [17, 18].Note that due to field renormalisation (31), M = λRφ2c/6 isrenormalisation group invariant rather than φ2c.Thus M is a physical quantityand is referred to as scalar condensate.

In the infinite volume limit (ai →∞) the9

effective potential has a global minimum for M = M0 ̸= 0, implying that the groundstate has a non-vanishing scalar condensate [18]. Furthermore, the minimum valueof the effective potential (vacuum energy density) is related to the scalar condensateby [18]U0 = −α144λ2R(φ2c)2 →−14β(λR)24⟨: φ4 :⟩,(38)which yields the correct scale anomaly at this level of approximation.In order to make renormalisation group invariance obvious, we remove the renor-malisation point dependence in (36) by subtracting U0 from the effective potentialU(M) (36).

Both the renormalisation point µ and the renormalised coupling λRdrop out, and we obtainU(M) = α2 M2(ln MM0−12) −αF3(M, a1 . .

. ap) .

(39)The effective potential U as a function of the scalar condensate M for one pair ofplates (p = 1) is shown in figure 1. In this case, the results are equivalent to thatof a finite temperature field theory if the inverse distance 1/a between the platesis identified with the temperature (in units of Boltzmann’s constant) [12].

Furtherresults of finite temperature φ4-theory are given in [18].For a large interplatedistance a (zero temperature), the continuum effective potential is obtained, and theeffective potential has a minimum at a nonvanishing value of the scalar condensate.At finite a, a second minimum at zero condensate M develops, which is referred toas the perturbative phase. At large a, this trivial phase is unstable, because the non-perturbative minimum has lower vacuum energy density.

Decreasing a (increasingtemperature), lowers the difference in the energy density between the perturbativeand non-perturbative phases. At a critical distance ac the non-trivial phase becomesdegenerate with the perturbative one (at M = 0).

If a is decreased further, the non-trivial phase becomes meta-stable and a first order phase transition to the trivialphase at M = 0 can occur, either by quantum or statistical fluctuations.3ResultsIn a free massless field theory (with p = 1) there is no intrinsic energy scale in com-petition with that of the interplate distance. This implies that the vacuum energydensity U0 scales as 1/a4 with the interplate distance from dimensional arguments.This scaling law was experimentally observed by Sparnaay in the case of QED [5].In the case of non-trivial φ4-theory an intrinsic energy scale is provided by the scalarcondensate.

Thus one expects deviations from the 1/a4 scaling law of the free theory.Such deviations might provide experimental access to properties of the non-trivialphase.10

3.1Near the infinite volume limitThe scalar condensate Mv at the minimum of the effective potential is given by thegap equationdUdM |M=Mv = Mv ln MvM0+ F2(Mv, a1 . .

. ap) = 0 .

(40)The vacuum energy density Uv is obtained by inserting Mv back into (39). For largeinterplate separations a2i ≫1/Mv the function Fǫ(Mv, a1 .

. .

ap) can be analyticallyestimated by noting that only terms with a single νi ̸= 0 and all others νj̸=i = 0contribute to the sum (28), i.e.Fǫ(M, a1 . .

. ap) ≈XiZ ∞0dssǫ e−sM e−a2i4s .

(41)A simple rescaling yieldsFǫ(M, a1 . .

. ap) =Xi( 4a2i)ǫ−1 fǫ(Mva2i4) ,fǫ(x) =Z ∞0dssǫ e−sx e−1s .

(42)After some technical manipulations, the functions fǫ(x) can be related to the mod-ified Bessel functions of the second kind, i.e.,fǫ(x) = 2 xǫ−12 Kǫ−1(2√x) ≈√π x2ǫ−34e−2√x ,(43)where the last approximate expression is just the asymptotic form of the Besselfunction for x →∞. Thus we have near the infinite volume limitF2 ≈M1/4√πXi(a2i4 )−3/4 e−√Mai ,F3 ≈M3/4√πXi(a2i4 )−5/4 e−√Mai .

(44)Solving (40) for the scalar condensate Mv in perturbation theory around M0 weobtain the change in the condensate due to the presence of the plates, i.e.Mv = M0 [1 −√πpXi=1(M0a2i4)−3/4 e−√M0ai + . .

. ] .

(45)There are two contributions to the variation of the vacuum energy densityUv(Mv(ai), ai) (39) with the interplate distances, one from a change of the scalarcondensate and one from the change of the effective potential U(M) via the functionF3. Equation (40) implies that a variation of the condensate does not change Uvin first order, and thus the leading contribution is from a change of F3.

Using theasymptotic form (44) for F3 one obtains1α∆Uv = 1α(Uv −U∞) ≈−√π M20pXi=1(M0a2i4)−5/4 e−√M0ai ,(46)11

where U∞is the vacuum energy density in the infinite volume limit. This is thedesired result: equation (46) gives the change of the vacuum energy density dueto boundary conditions.

In free field theory (and p = 1) it decays by the powerlaw ∼1/a4 (see (17)). In contrast, the energy density (46) of strongly interactingscalar modes decays exponentially (with a power law correction), the slope given bythe magnitude of the scalar condensate M0.

This implies that at least in principleone can decide by observing the dependence of the Casimir force on the interplatedistance, whether the theory is in a free or in a non-perturbative phase. In the lattercase, it is also possible to extract ground state properties, e.g.

the scalar condensate.Since in QED the 1/a4-power law was experimentally verified [5], the QED groundstate is trivial and e.g., has no photon condensate, an expected result since photonself-interactions are absent.3.2At the phase transition at small interplate distancesAs was seen in section 2.2 for one pair of plates (p = 1), the system undergoes a firstorder phase transition from the non-trivial vacuum to the perturbative vacuum,if the interplate distance becomes small enough. Numerical investigations of theeffective potential (39) at various distances ai show that the same effect holds forp > 1: if the box is small enough, a first order phase transition to the perturbativevaccum occurs.Equating the energy density U0 of the perturbative state at M = 0 to that of thenon-trivial phase (with non-zero condensate) at M = Mv we obtainM2v2 (ln MvM0−12) −F3(Mv, a1 .

. .

ap) + F3(0, a1 . .

. ap) = 0 ,(47)where the dependence of the scalar condensate Mv(a1 .

. .

ap) on the interplate dis-tances ai is implicitly given by (40). The set of equations (40, 47) defines a hyper-surface in the space spanned by the distances ai, which separates the non-trivialphase from the perturbative one.

Note that this transition line is given in terms ofrenormalisation group invariant (and therefore physical) quantities.For one pair of plates (p = 1), the formulation is equivalent to the finite-temperatureφ4-theory (identifying 1/a with temperature), and the phase transition at small dis-tance a is of same structure as that in the finite-temperature theory at high tem-perature. Due to this correspondence, the numerical value for the critical distancea(c) can be taken from [18]M0 a2(c) = 10.29134 .

. .

. (48)12

The ratio of the scalar condensate Mv at the transition point and the continuum(zero temperature) condensate M0 isMv(a(c)) / M0 = 0.9041 . .

. .

(49)Due to the first order nature of the phase transition, the scalar condensate has adiscontinuity at the transition point a(c) and is zero for smaller distances.For two pairsof plates the transition line between the two phases was obtainedby solving (40, 47) numerically.The result is presented in figure 2.Numericalinvestigations (cf. figure 4) suggest that the transition line is approximately givenby the equation1a2c 1+1a2c 2= M09.

(50)For p = 3 we have numerically checked, that the first order phase transition occurs,if the rectangular box is sufficiently small.3.3Boundary dependence of energy density and scalar con-densateFor given vacuum energy density Uv, equation (39) i.e.,M2v2 (ln MvM0−12) −F3(Mv, a1 . .

. ap) = Uv(51)with Mv defined by (40) yields the hypersurface of constant energy density in thespace spanned by {ai, i = 1 .

. .

p}. Comparing (51) with (47) it is easily seen that thephase separating surface is not a surface of constant energy density, implying thatthere are intersections between the two surfaces.

We expect that the hypersurfaceof constant energy density is continuous at the intersection, but not differentiabledue to the first order phase transition.Figure 3 shows the vacuum energy density for one pair of plates (p = 1) as a functionof 16/a2, when a is the interplate distance (or equivalently the inverse temperature).For large values of 16/a2 (small a), the perturbative phase is realised, and the 1/a4-scaling law is observed. For small values of 16/a2 (large a) the scalar theory is in thenon-trivial phase, and the energy density exponentially approaches the continuumvalue (see (46)) given by the scale anomaly (38).For two pairs of plates (p = 2), figure 4 shows lines of constant vacuum energydensity in the 16/a21 −16/a22 plane.

Also shown is the phase transition line (dashedcurve). The lines of constant energy density are continuous, but have a cusp at thefirst order phase transition point.13

We have also studied the hypersurfaces of constant scalar condensatein ai-space.For p = 1, this is equivalent investigating the temperature dependence of the scalarcondensate, and thus the results are given in [18]. For p = 2, the lines of constantcondensate (in units of continuum condensate M0) are presented in figure 5.

A lineof constant condensate is discontinuous at the phase transition line, and is zero in theperturbative phase. This behaviour is again due to the first order phase transition.4Discussion and concluding remarksWe have shown for φ4-theory constrained by a rectangular box that the non-trivialground state undergoes a first order phase transition to the perturbative vacuum, ifthe extension in at least one space-time direction becomes small enough.

For largeboxes the finite size corrections to the infinite volume limit are exponentially small.They are negligible, if the interplate distances are large compared with intrinsicscale provided by the (continuum) scalar condensate (i.e. ai√M0 ≫1).

On theother hand finite size effects become important for ai√M0 ≈1 and induces a phasetransition to a non-trivial vacuum.We believe that these properties are a common feature of a wide class of quantumfield theories.Indeed, an analogous situation is observed in lattice gauge theo-ries. Theoretical investigations show, that for high temperatures pure SU(N) latticegauge theory has a phase transition from a non-trivial (confining) ground state toa perturbative phase [27].

Numerical simulations of the SU(N) theory use a lat-tice with size ntn3, n ≫nt, which corresponds to a system with volume n3 andinverse temperature nt. Such a system shows two phases, a non-trivial phase forβ < βc(nt), and a deconfined phase for β > βc(nt) [28] (β = 2N/g2 with g theSU(N) coupling strength).

This compares well with our considerations as follows.The intrinsic scale of lattice gauge theories is provided by the string tension χ [29](or equivalently by the gauge field condensate [30] as in the continuum Yang-Millstheory). Our investigations suggest that a finite size phase transition occurs ifntn3 a4χ2 ≤1 ,(52)where a is the lattice spacing.

The string tension in units of the lattice spacing a4χ2,strongly depends on the inverse coupling strength β dictated by the renormalisationgroup. Numerical simulations [29] show that for fixed nt, n, a4χ2 decreases withincreasing β, implying that (52) is satisfied for β ≈βc, the coupling strength atwhich the phase transition occurs.In conclusion, we have studied φ4-theory in a renormalisation group invariant ap-proach inside a rectangular box consisting of p pairs of plates, at which the scalar14

modes satisfy periodic boundary conditions.We have further investigated theground state properties of the non-trivial phase affected by the geometrical con-straints. The dependence of the vacuum energy density and the scalar condensateon the interplate distances was studied in some detail.

In the non-trivial phase thevacuum energy density exponentially approaches the infinite volume limit, the decayrate given by the magnitude of the scalar condensate. This behaviour of the energydensity essentially differs form that of a free theory, where it scales according a 1/a4-power law.

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Figure captionsFigure 1:The effective potential as function of the scalar condensate at variousinterplate distances.Figure 2:The transition line separating the non-trivial phase and the perturbativephase, two pairs pf plates (p = 2) with distances a1 and a2, respectively.Figure 3:The vacuum energy density for one pair of plates (p = 1) as a functionof the interplate distance (inverse temperature) in units of 1/√M0.Figure 4:The lines of constant vacuum energy densityUvαM20 for two pairs of plates;a2i in units of the inverse scalar condensate 1/M0.Figure 5:The lines of constant scalar condensate M/M0 for two pairs of plates.18

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