On space-time noncommutative theories at finite temperature
aa r X i v : . [ h e p - t h ] J un On space-time noncommutative theories at finitetemperature
A. V. Strelchenko ∗ Dnepropetrovsk National University, 49050 Dnepropetrovsk, Ukraine
D. V. Vassilevich † Instituto de F´ısica, Universidade de S˜ao Paulo,Caixa Postal 66318 CEP 05315-970, S˜ao Paulo, S.P., Brazil
November 14, 2018
Abstract
We analyze renormalization and the high temperature expansion of theone-loop effective action of the space-time noncommutative φ theory byusing the zeta function regularization in the imaginary time formalism (i.e.,on S × R ). Interestingly enough, there are no mixed (non-planar) contri-butions to the counterterms as well as to the power-law high temperatureasymptotics. We also study the Wick rotation and formulate assumptionsunder which the real and imaginary time formalisms are equivalent. Since a number of review papers have been published recently (see [1]) it isnot necessary to repeat here the motivations for studying noncommutative (NC)field theories. Most of the previous previous works, see e.g. [2, 3, 4, 5] andreferences therein, on finite temperature NC theories analyzed the case of space-space noncommutativity (with very few exceptions [6, 7]). Indeed, the case ofspace-time noncommutativity is most problematic because of the difficulties withunitarity and causality which were discovered some years ago [8, 9, 10]. Thesedifficulties have not been completely resolved up to now. Space-time NC theorieswith compact dimensions exhibit an interesting phenomenon of discreteness oftime [11]. ∗ E.mail: alexstrelch(at)yahoo.com † On leave from V. A. Fock Institute of Physics, St. Petersburg University, Russia. E.mail: dmitry(at)dfn.if.usp.br φ on S × R to make sure that the theory which will be discussed laterdoes exist at least at the leading order of the loop expansion. We shall use thezeta-function regularization [12, 13] and the heat kernel technique [14, 15, 16]. Inthe context of an NC field theory the heat kernel expansion was first obtained forthe operators which contain only left or only right Moyal multiplications [17, 18].Such operators were, however, insufficient to deal with some physical applica-tions, like, for example, the φ theory. The heat kernel expansion for generalizedLaplacians containing both left and right Moyal multiplications was constructedin [19] on the Moyal plane and in [20] on the NC torus. Non-minimal operatorswere considered in [21]. We would also like to mention the calculations [22] ofthe heat kernel expansion in the NC φ model modified by an oscillator-typepotential.To avoid unnecessary technical complications we shall study exclusively thecase of pure space-time noncommutativity, i.e., we put to zero the NC param-eter with both indices in the spatial directions, θ jk = 0. We shall calculatethe heat kernel coefficients a n with n ≤
4. It will appear that the coefficients a and a look very similar to the commutative theory, but a is given by acomplicated non-local expression. Fortunately, odd-numbered heat kernel coeffi-cients do not contribute to one-loop divergences at four dimensions in the zeta-function regularization. The model will turn out to be one-loop renormalizablewith temperature-independent counterterms.Of course, we do not expect this model to be renormalizable at all loops.There are well-known problems related to the so-called UV/IR mixing [23] whichshould also be present in our case (though, maybe, in a relatively mild form sinceone of the NC directions is compact). To make the finite temperature NC φ renormalizable to all orders one should probably make it duality covariant [24]or use a bifermionic NC parameter [25].An approach to finite temperature theories on static backgrounds based onthe zeta-function regularization was developed long ago by Dowker and Kennedy[26]. In particular, they established relations between spectral functions of a3-dimensional operator which defines the spectrum of fluctuations and the hightemperature asymptotics of the free energy. In our case, due to the presenceof the space-time noncommutativity, such 3-dimensional operator becomes fre-quency dependent even on static backgrounds. Therefore, eigenfrequencies offluctuations are defined by a sort of a non-linear spectral problem. Fortunatelyfor us, a technique which allows to work with finite temperature characteristics ofthe theories leading to non-linear spectral problems has been developed relativelyrecently in the papers [27]. These papers were dealing with the thermodynam-ics of stationary but non-static space-times, but, after some modifications, the2pproach of [27] can be made suitable for space-time noncommutative theoriesas well. By making use of these methods we shall construct the spectral densityof states in the real-time formalism and express it through the heat kernel of afrequency dependent operator in three dimensions. Then, by using this spectraldensity, we shall demonstrate that the Wick rotation of the Euclidean free en-ergy gives the canonical free energy. To come to this conclusion we shall needtwo assumptions. First of all, we shall have to assume that the spectral densitybehaves ”nicely” as a function of complex frequencies. Although this assumptionis very hard to justify rigorously, we shall argue that the behavior of the spectraldensity must not be worse than in the commutative case, and we shall also sug-gest a consistency check based on the high temperature asymptotics. There isno canonical Hamiltonian in the space-time NC theories. Therefore, we have toassume that the eigenfrequncies of quantum fluctuations can replace one-particleenergies in thermal distributions. This assumptions cannot be derived from thefirst principles of quantization basing on the present knowledge on the subject,but we can turn the problem around: the very fact that the Wick rotation of theEuclidean free energy leads to a thermal distribution over the eigenfrequencessupports (a rather natural) guess that the eigenfrequences are the energies ofone-particle excitations. Let us stress, that the calculations we shall perform inthe Euclidean space do not depend on the assumptions described above.We shall also use the heat kernel methods to calculate the high temperatureasymptotics of the Euclidean effective action assuming that the background fieldis static. As in the case of the counterterms, there are non-planar contributions.The asymptotic expansion does not depend on the NC parameter (provided it isnon-zero) and looks very similar to the commutative case.This paper is organized as follows. In the next section we consider one-looprenormalization of NC φ on S × R . Sec. 3 is devoted to the Wick rotation.High temperature asymptotics of the effective action are calculated in sec. 4.Some concluding remarks are contained in sec. 5. S × R Let us consider a scalar φ model on NC S × R . The scalar field is periodic withrespect to the compact coordinate. We use the notations ( x µ ) = (¯ x, x ) = ( x i , x ),where x is a coordinate on S , 0 ≤ τ ≤ β . Similarly for the Fourier momentawe use k = (¯ k, k ), k = πlβ , l ∈ Z .The action reads S = 12 Z β dx Z R d ¯ x (cid:16) ( ∂ µ φ ) + m φ + g φ ⋆ (cid:17) , (1)3here the φ ⋆ = φ ⋆ φ ⋆ φ ⋆ φ . Star denotes the Moyal product f ⋆ f ( x ) = exp (cid:16) ı θ µν ∂ xµ ∂ yν (cid:17) f ( x ) f ( y ) | y = x . (2)To simplify the setup we assume that θ ij = 0, but some of θ j = 0, i.e. we havean Euclidean space-time noncommutativity.We wish to investigate quantum corrections to (1) by means of the backgroundfield method. To this end one has to split the field φ into a classical backgroundfield ϕ and quantum fluctuations, φ = ϕ + δϕ . The one-loop contribution tothe effective action is defined by the part of (1) which is quadratic in quantumfluctuations: S [ ϕ, δϕ ] = 12 Z β dx Z R d ¯ x δϕ ( D + m ) δϕ, (3)where the operator D is of the form (cf. [28, 19]) : D = − ( ∂ µ ∂ µ + E ) , (4)with E = − g L ( ϕ ⋆ ϕ ) + R ( ϕ ⋆ ϕ ) + L ( ϕ ) R ( ϕ )) . (5)The one-loop effective action can be formally written as W = 12 ln det( D + m ) . (6)This equation still has to be regularized. To make use of the zeta-function regu-larization we have to define the heat kernel K ( t, D ) = Tr (cid:0) e − tD − e tD (cid:1) (7)and the zeta function ζ ( s, D + m ) = Tr (cid:0) ( D + m ) − s − ( D + m ) − s (cid:1) . (8)Here Tr is the L trace. In both cases we subtracted the parts corresponding tofree fields with D = − ∂ µ ∂ µ to avoid volume divergences.The regularized one-loop effective action is defined as W s = − µ s Z ∞ dtt − s e − tm K ( t, D ) = − µ s Γ( s ) ζ ( s, D + m ) , (9)where s is a regularization parameter, µ is a constant of the dimension of massintroduced to keep proper dimension of the effective action. The regularization A better name used in mathematics for this object is the heat trace, but here we use theterminology more common in physics.
4s removed in the limit s →
0. At s = 0 the gamma-function has a pole, so thatnear s = 0 W s = − (cid:18) s − γ E + ln µ (cid:19) ζ (0 , D + m ) − ζ ′ (0 , D + m ) , (10)where γ E is the Euler constant.Let us assume that there is an asymptotic expansion of the heat kernel as t → +0 K ( t, D ) = ∞ X n =1 t ( n − / a n ( D ) . (11)Such an expansion exists usually (but not always) in the commutative case. OnNC S × R the existence of (11) will be demonstrated in sec. 2.2. For a Laplacetype operator on a commutative manifold all odd-numbered heat kernel coeffi-cients a k − vanish. (They are typical boundary effects). As we shall see below,on NC S × R the coefficient a = 0. The coefficient a disappears due to thesubtraction of the free-space contribution in (7).The pole part of W s can be now expressed through the heat kernel coefficients. ζ (0 , D + m ) = − m a ( D ) + a ( D ) . (12)Note, that odd-numbered heat kernel coefficients a p − ( D ) do note contribute tothe divergences of W s . S × R Let us consider the operator D = − ( ∂ µ + E ) , E = L ( l ) + R ( r ) + L ( l ) R ( r ) (13)on S × R . This operator is slightly more general than the one in (4). Thepotential term (5) is reproduced by the choice l = r = − g ϕ ⋆ ϕ , l = − r = r g ϕ . (14)We are interested in the asymptotics of the heat trace (7) as t → +0. To calculatethe trace we, as usual, sandwich the operator between two normalized planewaves , and integrate over the momenta and over the manifold M = S × R . K ( t ; D ) = 1 β (2 π ) Σ Z dk Z M d x e − ıkx (cid:0) e − tD − e tD (cid:1) e ıkx , (15) Although we are working with a real field, it is more convenient to use complex plane wavesinstead of real functions sin( kx ) and cos( kx ). For a complex field we would have a coefficientof 1 instead of 1 / D with (14) is real, this is the onlydifference. Z dk ≡ X k Z d ¯ k (16)with k = 2 πn/β , n ∈ Z . To evaluate the asymptotic expansion of (15) at t → +0one has to extract the factor e − tk . K ( t, D ) = 1 β (2 π ) Z d x Σ Z dke − tk ×h exp (cid:0) t (cid:0) ( ∂ − ik ) + 2 ik µ ( ∂ µ − ik µ ) + E (cid:1)(cid:1) − i k , (17)where we defined h F i k ≡ e − ıkx ⋆ F e ıkx (18)for any operator F . Next one has to expand the exponential in (17) in a powerseries in E and ( ∂ − ık ). As we shall see below, only a finite number of termsin this expansion contribute to any finite order of t in the t → +0 asymptoticexpansion of the heat kernel. We push all ( ∂ − ık ) to the right until they hit e ıkx and disappear. K ( t, D ) = 1 β (2 π ) Z d x Σ Z dke − tk (cid:28) tE + t ∂ µ , [ ∂ µ , E ]] + E + 2 ık µ [ ∂ µ , E ]) − t k µ k ν [ ∂ µ , [ ∂ ν , E ]] + . . . (cid:29) k . (19)We kept in this equation all the terms which may contribute to a n with n ≤
4. Inthe commutative case all total derivatives as well as all term linear in k vanish.In the NC case this is less obvious because of the non-locality, so that we keptalso such terms. The commutator of ∂ µ with E is a multiplication operator,e.g., [ ∂ µ , L ( l )] = L ( ∂ µ l ), [ ∂ µ , L ( l ) R ( r )] = L ( ∂ µ l ) R ( r ) + L ( l ) R ( ∂ µ r ). Therefore,eq. (19) contains multiplication operators of two different sorts: the ones withonly left or only right Moyal multiplications, and the ones containing products ofleft and right Moyal multiplications. The terms of different sorts will be treateddifferently .The terms with one type of the multiplications are easy. We shall call suchterms planar borrowing the terminology from the approach based on Feynmandiagrams. They can be evaluated in the same way as in [17, 18]. Because of theidentities Z d x h R ( r ) i k = Z d x r ( x ) , Z d x h L ( l ) i k = Z d x l ( x ) (20) Formally R ( r ) = L (1) R ( r ), but a constant function does not belong to C ∞ ( S × R ) sinceit does not satisfy the fall-off conditions. Consequently, the two sorts of the term discussedabove indeed lead to quite different asymptotics at t → +0. E and E terms in (19) contribute. It remains then to evaluate theasymptotics of the integral1 β (2 π ) Σ Z dk e − tk = (4 πt ) − + e.s.t. , (21)where e.s.t. denotes exponentially small terms, to obtain a planar2 ( D ) = (4 π ) − Z d x ( l + r ) , (22) a planar4 ( D ) = (4 π ) − Z d x
12 ( l + r ) . (23)Non-planar (mixed) contributions require considerably more work. The typi-cal term reads T ( l, r ) = 1 β (2 π ) Z d x Σ Z dke − tk h L ( l ) R ( r ) i k (24)with some functions r ( x ) and l ( x ). For example, taking l = l and r = tr theexpression (24) reproduces the first term ( tE ) in (19). Let us expand r ( x ) and l ( x ) in the Fourier integrals r ( x ) = 1 β / (2 π ) / Σ Z dq ˜ r ( q ) e ıqx ,l ( x ) = 1 β / (2 π ) / Σ Z dq ′ ˜ l ( q ′ ) e ıq ′ x . (25)Then h L ( l ) R ( r ) i k = 1 β (2 π ) Σ Z dq Σ Z dq ′ ˜ r ( q )˜ l ( q ′ ) e ı ( q + q ′ ) x e − ı k ∧ ( q − q ′ ) − ı ( q ′ − k ) ∧ ( q + k ) , (26)where k ∧ q ≡ θ µν k µ q ν . (27)Next we substitute (26) in (24) and integrate over x and q ′ to obtain T ( l, r ) = 1 β (2 π ) Σ Z dk Σ Z dqe − tk ˜ l ( − q ) ˜ r ( q ) e − ık ∧ q . (28)In our case k ∧ q = θ i ( k q i − k i q ).Next we study the integral over k . The sum over k is treated with the helpof the Poisson formula X n ∈ Z f (2 πn ) = 12 π X n ∈ Z Z ∞−∞ f ( p ) e − ınp dp. (29)7e apply this formula to the sum X k exp( − tk − ıθ j k q j ) , (30)which corresponds to the choice f ( p ) = exp (cid:18) − tp β − ıθ j q j pβ (cid:19) (31)in (29). The sum (30) is transformed to (after changing the integration variable y = p/β ) β π X n ∈ Z Z ∞−∞ dy exp( − ty − ıy ( θ j q j + βn )) , = β π X n ∈ Z r πt exp (cid:18) − ( θ j q j + βn ) t (cid:19) . (32)The integral over k j is Gaussian and can be easily performed. We arrive at T ( l, r ) = 1(4 πt ) Σ Z dq X n exp (cid:18) − | θ | q + ( θ j q j + βn ) t (cid:19) h ( q ) , (33)where h ( q ) ≡ ˜ l ( − q ) ˜ r ( q ) , | θ | ≡ θ j θ j . (34)In eq. (33) one can still put | θ | = 0 thus returning to the commutative case.The limit | θ | → t → | θ | 6 = 0. Obviously, all terms in the sum over q areexponentially small as t → +0 except for q = 0. T ( l, r ) = 1(4 πt ) Z d ¯ q X n exp (cid:18) − ( θ j q j + βn ) t (cid:19) h (0 , ¯ q ) + e.s.t. (35)Let us define two projectorsΠ ij k = θ i θ j | θ | , Π ij ⊥ = δ ij − Π ij k (36)and split ¯ q into the parts which are parallel and perpendicular to θ j : q k =Π k ¯ q , q ⊥ = Π ⊥ ¯ q . Then d ¯ q = dq k d q ⊥ , and ( θ j q j + βn ) = ( | θ | q k + βn ) . Theasymptotics of the integral over q k can be calculated by the saddle-point method.For each n there is one critical point of the integrand corresponding to q k = q ( n ) k ≡ βn/ | θ | . We expand h (0 , q k , q ⊥ ) about these critical points and take the integralover q k to obtain T ( l, r ) = 1 | θ | (4 πt ) / X n ∈ Z Z d q ⊥ (cid:18) h (0 , q ( n ) k , q ⊥ ) + t | θ | h ′′ (0 , q ( n ) k , q ⊥ ) + . . . (cid:19) , (37)where prime denotes derivative with respect to q k . This completes the calculationof small t asymptotics for T ( l, r ). Since both l ( x ) and r ( x ) are supposed to besmooth, their Fourier components ˜ l ( q ) and ˜ r ( q ) fall off faster than any powerat large momenta, and each term in the asymptotic expansion is given by aconvergent sum and a convergent integral.The expression (37) is already enough to calculate mixed (non-planar) contri-butions to the heat kernel expansions from the terms inside the brackets in (19)which do not contain k . (We shall do this in a moment). Regarding the termswhich do contain the momentum k , for our purposes it is enough to evaluate thepower of t appearing in front of such terms. One can easily trace which modifi-cations appear in the calculations (24) - (37) due to the presence of a polynomialof k µ . The result is: (i) we still have an expansion in t / , (ii) the terms with k do not contribute to the heat kernel coefficients a n with n ≤
4. In other words,the only relevant mixed heat kernel coefficient is generated by the first term inthe brackets in (19), and it reads a mixed3 ( D ) = 1 | θ | (4 π ) / X n ∈ Z Z d q ⊥ ˜ l (0 , − q ( n ) k , − q ⊥ )˜ r (0 , q ( n ) k , q ⊥ ) , (38)where we substituted the fields appearing in E (see eq. (13)). Note, that thisexpression is divergent in the commutative limit | θ | →
0. The coefficient a mixed3 is highly non-local. The structure of (38), especially the sum over n , remindsus of the heat kernel coefficients on NC torus for a rational NC parameter [20].In this latter case there is a simple geometric interpretation in terms of periodicprojections [29]. No such interpretation is known for the present case of S × R .However, some similarities can be found to the works [11] discussing discretizationof the coordinates which do not commute with a compact coordinate. Let us return to our particular model (1). First we summarize the results of theprevious subsection and re-express the heat kernel coefficients a n = a planar n + a mixed n
9n terms of the background field ϕ by means of (14): a ( D ) = − g π Z d xϕ , (39) a ( D ) = − g | θ | (4 π ) / X n ∈ Z Z d q ⊥ ˜ ϕ (0 , − q ( n ) k , − q ⊥ ) ˜ ϕ (0 , q ( n ) k , q ⊥ ) (40) a ( D ) = 116 π g Z d xϕ ⋆ , (41)where tilde is used again to denote the Fourier components.Next we substitute (39) - (41) in (10) and (12) to the pole part of the regu-larized effective action W pole s = − s Z d x (cid:18) g π m ϕ + 116 π g ϕ ⋆ (cid:19) . (42)This divergent part of the effective action can be cancelled by an infinite renor-malization of couplings in (1) δm = gm π s , δg = g π s . (43)There can be, of course, also some finite renormalization which we do not discusshere. Our main physical observation in this subsection the renormalization (43)does not depend on the temperature 1 /β .Here some more comments are in order. It is a very attractive feature ofthe zeta function regularization that the non-planar non-local coefficient a ( D )does not affect the counterterms. This coefficient will, however, contribute atsome other places, like the large mass expansion of the one-loop effective action(see, e.g., [16]). Moreover, a ( D ) can lead to troubles in different regularizationschemes. For example, if one uses the proper-time cut-off at some scale Λ definingthe regularized effective action by W Λ = − Z ∞ / Λ dtt e − tm K ( t, D ) , (44)the coefficient a generates a linear divergence ∝ Λ a ( D ), which has no classicalcounterpart and cannot be renormalized away in the standard approach. Thereis a subtraction scheme (that was used in quantum field theory on curved back-ground [30] and in Casimir energy calculations [31]) which prescribes to subtractall contributions from several leading heat kernel coefficients, including a ( D ) infour dimensions. In the case of two-dimensional scalar theories this heat kernelsubtraction scheme is equivalent to usual renormalization with the “no-tadpole”normalization condition [32]. In the present case the heat kernel subtraction is,obviously, not equivalent to the charge and mass renormalizations given by (43).10e restricted ourselves to the case of pure space-time noncommutativity θ ij =0. However, one can try to make an educated guess on what happens for ageneric non-degenerate θ µν . By comparing the heat kernel expansion obtainedabove with that on NC torus [20] and on NC plane with a non-degenerate θ µν [19] we can derive (rather unrigorously) the following rule: the presence of anon-compact NC dimension increases the number of the first non-trivial field-dependent non-planar (mixed) heat kernel coefficient by one as compared to thefirst non-trivial field-dependent coefficient in the commutative case. Indeed, inthe commutative case the first such coefficient is a . On the NC torus [20] (nonon-compact dimensions) the first field-dependent mixed heat kernel coefficient isalso a . For the geometry studied in this paper (one non-compact NC dimension)this is a . On an n -dimensional NC plane with a non-degenerate θ µν the firstcoefficient of interest is a n +2 [19]. We can expect therefore that the first mixedcoefficient on S × R with a non-degenerate θ µν (three compact NC dimensions)will be a . Such coefficient does not contribute to one loop divergence neitherin the zeta-function regularization, nor in the proper-time cut-off scheme. Thusthe situation in the generic case may be expected to better that in the case ofa degenerate θ µν discussed above. A similar conclusion has been made for theMoyal plane in [28].As we have already mentioned above, the counterterms do not depend onthe temperature. However, if one does the calculations directly on the zero-temperature manifold R , there appears problems for a degenerate NC parameter[28]. Perhaps, compactification of one of the NC directions is a proper way toregularize these problems away. The methods which allow to make correspondence between imaginary and realtime formalisms in the case of frequency-dependent Hamiltonians were suggestedin [27] and developed further in [33]. Here we briefly outline these methods anddiscuss the peculiarities of their application to noncommutative theories. Fromnow on we work with static background fields, ∂ ϕ = ∂ ϕ = 0. Let us consider a Minkowski space counterpart of the action (1). Our rules forthe continuation between Euclidean and Minkowski signatures read ∂ → ı∂ and θ j → − ıθ j , where θ j is real, and θ j ∂ corresponds to θ j ∂ . We have,therefore, a real NC parameter in the Moyal product on both Euclidean andMinkowski spaces. These rules were applied, e.g., in [9], and they follow alsofrom the requirement of reflection positivity [34]. As we shall see below, theserules also ensure consistency between the expressions for the free energy defined11n imaginary and real time formalisms.The wave equation for quantum fluctuations ψ ( x ) over a static backgroundreads (cid:16) − ∂ + ∂ j − m − g L ( ϕ ) + R ( ϕ ) + L ( ϕ ) R ( ϕ )) (cid:17) ψ ( x ) = 0 . (45)The wave operator in (45) commutes with ∂ . Consequently, one can look for thesolutions ψ ω whose time dependence is described by ψ ω ( x ) ∼ e iωx . They satisfythe equation ( P ( ω ) + m ) ψ ω = ω ψ ω , (46)where P ( ω ) = − ∂ j + V ( ω ) , V ( ω ) = g ϕ + ϕ − + ϕ + ϕ − ) (47)and ϕ ± ( x j ) = ϕ (cid:18) x j ± θ j ω (cid:19) . (48)Here we used the fact that left (right) Moyal multiplication of a function of x j by exp( iωx ) is equivalent to a shift of the argument.From now on we consider the case of positive coupling g only. Then thepotential V ( ω ) is non-negative, V = ( g/ ϕ + ϕ − + ( ϕ + + ϕ − ) ) ≥ P ( λ ) + m ) ψ ν,λ = ν ψ ν,λ . (49)Obviously, the functions ψ ω,ω solve the equation (46).Our next step differs from that in [27]. Let us restrict λ to λ ≤ λ forsome λ and put the system in a three dimensional box with periodic boundaryconditions. Let us suppose that the size of the box is ≫ θλ , so that ϕ + and ϕ − are localized far away from the boundaries. In this case, the spectrum of theregularized problem can be considered as an approximation to the spectrum ofthe initial problem for the whole range of ν . Later we shall remove the box, andthe restriction λ ≤ λ will become irrelevant. In the box, the spectrum of ν in(49) becomes discrete, but, for a sufficiently large box, the spacing is small. Theeigenvalues ν N ( λ ) depend smoothly on λ not greater than λ , and we can definethe density of states as dn ( ν, λ ) d ( ν ) = 12 ν dn ( ν, λ ) dν = X N δ ( ν − ν N ( λ )) , (50)which can be used to calculate spectral functions of ˜ P ( λ ), where tilde reminds usthat we are working with a finite-volume problem. For example, f Tr( e − t ( ˜ P ( λ )+ m ) = Z ∞ m dn ( ν, λ ) dν e − tν dν . (51)12ere f Tr denotes the L trace in the box. The potential V is non-negative. Con-sequently, there are no eigenvalues below m .The eigenvalues ω N of the initial problem (46) in this discretized setting ap-pear when the line ν = λ intersects ν N ( λ ). We can define the density of theeigenfrequencies ω N by the formula dn ( ω ) d ( ω ) = X N δ ( ω − ω N ) . (52)Next, we would like to relate this density to (50). This can be done by calculatingderivative of the arguments of the delta function taken for ω = λ = ν . We obtain dn ( ω ) d ( ω ) = d ˆ n ( ω, ω ) d ( ω ) , (53)where d ˆ n ( ν, λ ) d ( ν ) = X N (cid:18) − d ( ν N ) d ( λ ) (cid:19) δ ( ν − ν N ) . (54)This density admits an interpretation in terms of the heat kernel f Tr " − λ d ˜ P ( λ ) dλ ! e − t ( ˜ P ( λ )+ m ) = (cid:18) λt ddλ (cid:19) f Tr( e − t ( ˜ P ( λ )+ m ) )= Z ∞ m d ˆ n ( ν, λ ) dν e − tν dν . (55)Next we remove the box. Most of the quantities discussed above are divergentin the infinite volume limit. In order to remove these divergences we subtract thespectral densities corresponding the the free operator ˜ P + m with ˜ P = − ∂ j (notto be confused with ˜ P (0)). Then we perform the infinite volume limit. The limitsof subtracted densities dn ( ω ) /dω , dn ( ν, λ ) /dλ and d ˆ n ( ν, λ ) /dλ will be denoted by ρ ( ω ), ρ ( ν, λ ) and ̺ ( ν, λ ), respectively. The following relation holds in this limit:Tr (cid:16) e − t ( P ( λ )+ m ) (cid:17) sub = Z ∞ m dω ρ ( ω ; λ ) e − tω , (56)where Tr is the L trace on R andTr (cid:16) e − t ( P ( λ )+ m ) (cid:17) sub ≡ Tr (cid:16) e − t ( P ( λ )+ m ) − e − t ( − ∂ j + m ) (cid:17) . (57)We also have the relation (cid:18) λt ddλ (cid:19) Tr (cid:16) e − t ( P ( λ )+ m ) (cid:17) sub = Z ∞ m ̺ ( ν ; λ ) e − tν dν , (58)13hich, together with (56), yields ̺ ( ω ; λ ) = ρ ( ω ; λ ) + ωλ Z ωm ∂ λ ρ ( σ ; λ ) dσ . (59)To derive this formula one has to integrate by parts. Vanishing of the boundaryterms is established by using the same arguments as in [27]. An infinite volumecounterpart of (53) reads ρ ( ω ) = ̺ ( ω ; ω ) . (60)An independent calculation of the spectral densities is a very hard problem.We shall view the equation (56) as a definition of the subtracted spectral density ρ ( ν ; λ ) through the heat kernel (an explicit formula involves the inverse Laplacetransform). The other spectral densities ̺ ( ν ; λ ) and ρ ( ω ) are then defined through(59) and (60).Relations similar to (56), (58), (59) and (60) were originally obtained in [27]for a different class of frequency dependent operators and by a somewhat differentmethod. In this section we show that the Wick rotation of the free energy F definedthrough the Euclidean effective action coincides with the canonical free energy F C . The methods we use are borrowed from [27], but there are some subtle pointsrelated to specific features of NC theories. By definition, W ( β ) = β ( F ( β ) + E ) , (61)where E is the energy of vacuum fluctuations.Our renormalization prescription (43) ia equivalent to the (minimal) subtrac-tion of the pole term (42) in (10). Therefore, the renormalized one-loop effectiveaction reads W = − dds | s =0 (cid:0) ˜ µ s ζ ( s, D + m ) (cid:1) , (62)where ˜ µ := µ e − γ E . On a static background one can separate the frequency sumfrom the L ( R ) trace and rewrite the zeta function as ζ ( s, D + m ) = X l Tr (cid:0) ( ω l + m + P ( ω l )) − s − ( ω l + m − ∂ j ) − s (cid:1) = X l Z ∞ m dν ρ E ( ν ; ω l )( ω l + ν ) − s . (63) ω l = 2 πl/β . The spectral density ρ E is defined for the Euclidean space NCparameter θ j . It is related to the real-time spectral density by the formula ρ E ( ν ; ω | θ j ) = ρ ( ν ; ıω | − iθ j ) (64)14ccording to the rules which we have discussed at the beginning of sec. 3.1. Wehave already mentioned that the Wick rotation leaves the combination ωθ andthe potential V invariant. Therefore, both densities coincide as functions of theirarguments ν and ω . However, we shall keep the subscript E to avoid confusion,but shall drop θ from the notations for the sake of brevity. Next we use theformula X l f ( ω l ) = β πı I C cot (cid:18) βz (cid:19) f ( z ) dz (65)with the contour C consisting of two parts, C + running from ıǫ + ∞ to ıǫ − ∞ and C − running from − ıǫ − ∞ to − ıǫ + ∞ , to rewrite the frequency sum as anintegral. Then, by using the symmetry of the integrand with respect to reflectionsof z we replace the integral over C by twice the integral over C + alone. Finally,we apply the identity cot (cid:18) βz (cid:19) = 2 β ddz ln(1 − e ıβz ) − ı (66)to arrive at the result ζ ( s, D + m ) = βζ ( s, D + m ) + ζ T ( s, D + m ) , (67)where ζ ( s, D + m ) = 1 π ∞ Z m dν ∞ Z ρ E ( ν ; z )( ν + z ) − s dz , (68) ζ T ( s, D + m ) = 1 πı ∞ Z m dν I C + dz (cid:20) ddz ln(1 − e ıβz ) (cid:21) ρ E ( ν ; z )( ν + z ) − s . (69)In commutative theories [27], the function ζ T , which vanishes at zero temperature,represents the purely thermal part, while ζ is responsible for the vacuum energy.In space-time NC theories there is no good definition of the canonical Hamiltonianand of the energy. Therefore, we have no other choice than to accept the sameidentities as in the commutative case, namely F ( β ) = − β dds | s =0 ˜ µ s ζ T ( s, D + m ) , (70) E = − dds | s =0 ˜ µ s ζ ( s, D + m ) . (71)Actually, the definition of E is a rather natural one since it coincides with therenormalized Euclidean one-loop effective action on R . However, as we havealready mentioned in sec. 2.3 the renormalization in NC theories depends cruciallyon the number of compact dimension. Therefore, if one does the renormalization15irectly in R , one may need the counterterms which differ from (43) obtainedon S × R .From now on we concentrate exclusively on F T ( β ) and ζ T . We integrate byparts over z to obtain ζ T ( s ) = − πı ∞ Z m dν I C + dz ln(1 − e ıβz ) (cid:20) ∂ z ρ E ( ν ; z )( z + ν ) s − zs ρ E ( ν ; z )( z + ν ) s +1 (cid:21) . (72)To ensure the absence of the boundary terms we have to deform the contour C + by moving its’ ends up in the complex plane, so that e ıβz provides the necessarydamping of the integrand. We discuss the conditions on ρ E which make suchdeformations of the contour legitimate below. The integration by parts over ν inthe first term in the square brackets in (72) yields ζ T ( s ) = sπı ∞ Z m dν I C + dz ln(1 − e ıβz ) 2 z ( z + ν ) s +1 ̺ E ( ν ; z ) , (73)where ̺ E ( ν ; z ) = ρ E ( ν ; z ) − νz Z νm ∂ z ρ E ( σ ; z ) dσ . (74)The right hand side of (73) is proportional to s . To estimate the derivative ∂ s at s = 0 in (70) one can put s = 0 in the rest of the expression and use the Cauchytheorem after closing the contour in the upper part of the complex plane. Theresult is then given by the residue at z = iν . Next we make the Wick rotationof the NC parameter, so that ρ E ( σ ; iν ) becomes ρ ( σ ; ν ), and ̺ E ( ν ; iν ) becomes ρ ( ν ; ν ) = ρ ( ν ) (cf. eqs. (59) and (60)). Consequently, the Euclidean free energyis given by the equation F ( β ) = 1 β ∞ Z m dν ρ ( ν ) ln(1 − e − βν ) , (75)which coincides with the canonical definition of the free energy F C .The equality F = F C is the main result of this section. To derive it we in-tegrated by parts and deformed the contour C + . The integration by part over ν is a safe operation, since for any fixed z the spectral density ρ E ( ν, z ) corre-sponds to the Laplace operator with a smooth potential. The absence of theboundary terms can be then demonstrated by standard arguments [27] based onthe heat kernel expansion. The deformations of the contour are more tricky. Tojustify this procedure and application of the Cauchy theorem one has to assumethat ρ E ( ν, z ) can be analytically continued to the upper half-plane as an entirefunction of z . A rigorous proof of this assumption is hardly possible even in16ore tractable cases of stationary commutative space-times [27]. We may argue,however, that this assumption is plausible. Consider pure imaginary values of z = iκ . All deformations of the contour are done before the Wick rotation of theNC parameter θ . Therefore, ϕ ± becomes complex, and ϕ + = ϕ ∗− . The potential V ( iκ ) remains real and positive. The background field ϕ is assumed to fall offfaster than any power of the coordinates in real directions to ensure the existenceof the heat kernel expansion. Such fields typically grow in imaginary directions(one can consider ϕ ∼ e − cx as an example). Large positive potentials tend todiminish the spectral density thus preventing it from the blow-up behavior. Itseems therefore, that the spectral density in our case should not behave worsethan the spectral density in the commutative case. Another argument in favorof our assumption will be given at the end of the next section.The free energy (75) is expressed through a thermal distribution over theeigenfrequencies. In the absence of a well-defined Hamiltonian it is not guaranteedthat this is the same as a thermal distribution of one-particle energies. This isa known problem of space-time NC theories which is beyond the scope of thispaper. As in the previous section we rewrite the regularized one-loop effective action (9)on a static background in the form W s = − µ s Γ( s ) X ω Tr ( ω + P ( ω ) + m ) − s sub , (76)where the sum over the Matsubara frequencies is separated from the trace overthe L functions on R . As usual, we subtracted the free space contributionscorresponding to ϕ = 0 in P ( ω ) (which is indicated by the subscript ”sub” in(76)). We remind that ω = 2 πl/β , l ∈ Z . To evaluate the high temperature(small β ) asymptotics of W s we split the sum in (76) in two parts, W s = W l =0 s + W l =0 s , (77)which will be treated separately.We start with W l =0 s which reads W l =0 s = − µ s Γ( s ) Tr ( P (0) + m ) − s sub = − µ s Γ( s ) ζ ( s, P (0) + m ) . (78)(The subtraction of free space contributions is included in our definition of thezeta function, cf. (8)). For each given ω the operator P ( ω ) is a three-dimensionalLaplace operator with a scalar potential. All effects of the noncommutativity areencoded in the form of this potential. Therefore, as for all Laplace type operators17n R , the zeta function in (78) vanishes at s = 0 making W l =0 s finite. We canimmediately take the limit s → W l =0 = − ζ ′ ( P (0) + m ) . (79)In the rest of the frequency sum we first use an integral representation for thezeta function W l =0 s = − µ s X ω =0 Z ∞ dt t s − Tr (cid:16) e − t ( ω + m + P ( ω )) (cid:17) sub (80)and then use a trick similar to the one employed in the previous section. Namely,we replace the operator in the exponential on the right hand side of (80) by ω + m + P ( λ ), expand each of the terms under the frequency sum in asymptoticseries at ω → ∞ keeping λ fixed, and then put λ = ω . The result of this procedurereads W l =0 s = − µ s X ω =0 ∞ X n =2 Z ∞ dt t s − t n − e − tω a n ( P ( ω ) + m ) sub = − µ s X ω =0 ∞ X n =2 | ω | − n − s Γ (cid:18) n −
32 + s (cid:19) a n ( P ( ω ) + m ) sub . (81)Some comments are in order. Here we used again the fact that ω + m + P ( λ ) fora fixed ω is just a usual Laplace type operator in three dimensions. The large ω expansion of the heat trace in (80) is therefore standard and, as well as the usuallarge mass expansion is defined by the heat kernel coefficients (see, e.g., [16]). Ona manifold without boundary an asymptotic expansion (11) with the replacement D → P + m exists, and only even numbers n appear. The coefficient a vanishesdue to the subtraction, so that the sum in (81) starts with n = 2.Now, we have to study the behavior of a n ( P ( ω ) + m ) sub at large ω . Theseheat kernel coefficients are integrals over R of polynomials constructed from thepotential V ( ω ) and its derivatives. We can present them as a n ( P ( ω ) + m ) sub = a n ( P + m ) planarsub + a n ( P ( ω ) + m ) mixed , (82)where the first (planar) contribution contains all terms which are the products ofeither ϕ + and its derivatives only, or of ϕ − and its derivatives only (but not theproducts of ϕ + and ϕ − ). The rest is collected in the second (mixed) contribution.Obviously, no subtraction for the mixed heat kernel coefficient is needed. Due tothe translation invariance of the integral over R , the planar coefficient does notdepend on ω . E.g., R d xϕ = R d xϕ . Therefore, we drop ω from the notation.First, let us consider the mixed contributions to (81). We assumed thatthe background field ϕ belongs to C ∞ ( S × R ). Therefore, it should vanish18xponentially fast at large distances. Since each term in a n ( P ( ω ) + m ) mixed contains a product of at least one ϕ + with at least one ϕ − , it should be oforder C e − C | ωθ | for large ω , where C and C are some constants. C is positiveand characterizes the fall-off of ϕ at large distances. C depends on n , on theamplitude of ϕ , and on the functional form of a n . Up to an inessential overallconstant the contribution of a mixed coefficient to (81) can be estimated as ∼ X ω =0 | ω | − n e − C | ωθ | ∼ ∞ X l =1 β n − l − n exp (cid:18) − πC l | θ | β (cid:19) (83)(this sum is obviously convergent, so that one can remove the regularizationparameter). If β is small enough, namely β ≪ C | θ | , all terms in the sum (83)are strongly suppressed, and the value of the sum can be well approximated bythe first term ∼ β n − exp (cid:18) − πC | θ | β (cid:19) . (84)We conclude that the contributions of the mixed terms are exponentially smalland can be neglected in the high temperature expansion of the effective action.Since the planar heat kernel coefficients do not depend on ω we are ready toevaluate their contribution to (81) by using precisely the same procedure as inDowker and Kennedy [26]. W l =0 s = − µ s ∞ X n =2 ∞ X l =1 Γ (cid:18) n −
32 + s (cid:19) l − n − s a n ( P + m ) planarsub (cid:18) β π (cid:19) n − s = − µ s ∞ X n =2 Γ (cid:18) n −
32 + s (cid:19) ζ R (2 s + n − a n ( P + m ) planarsub (cid:18) β π (cid:19) n − s (85)with ζ R being the Riemann zeta function. We remind that the index n in (85) iseven. The only divergence in (85) is a pole in ζ R for n = 4. The correspondingterm near s = 0 reads12 a ( P + m ) planarsub β (4 π ) / (cid:20) − s − γ E − (cid:18) µβ π (cid:19)(cid:21) . (86)On static backgrounds there is a useful formula which relates planar heat kernelcoefficients of D and P , a n ( D + m ) planarsub = β (4 π ) / a n ( P + m ) planarsub . (87)This formula follows from the analysis of the planar heat kernel coefficients pre-sented in sec. 2.2 and general formulae for the heat kernel expansion of Laplace If one imposes a stronger restriction on the background requiring that ϕ is of compactsupport, then the mixed terms vanish identically above certain temperature. β appears due to the integration ofa constant function over the Euclidean time, and (4 π ) / comes from differentprefactors in the heat kernel coefficients in 3 and 4 dimensions. In particular, a ( D + m ) planarsub = − m a ( D ) + a ( D ) = β/ (4 π ) / a ( P + m ) planarsub (let us re-mind that mixed a ( D ) and a ( D ) vanish). From (10) and (12) we see that thedivergence in the Euclidean effective is reproduced. This divergence is then re-moved by the renormalization of couplings (43). After the renormalization, wecollect all contributions to the effective action to obtain our final result for thehigh temperature expansion of the renormalized effective action W = − π / β a ( P + m ) planarsub − ζ ′ ( P (0) + m ) − a ( P + m ) planarsub β (4 π ) / (cid:20) γ E + 2 ln (cid:18) µβ π (cid:19)(cid:21) (88) − ∞ X n =6 Γ (cid:18) n − (cid:19) ζ R ( n − a n ( P + m ) planarsub (cid:18) β π (cid:19) n − . It is instructive to compare the expansion (88) to the one in the commutativecase obtained by Dowker and Kennedy [26] (note, that the normalization of theheat kernel coefficients used in that paper differs from ours). We see that the ζ ′ term is the same in both cases. The terms proportional to the heat kernelcoefficients for the commutative case can be obtained from the expansion aboveby means of the replacement a n ( P + m ) planarsub → a n ( P (0) + m ) sub . (In bothcases subtraction of the free space contribution means simply deleting the highestpower of m in standard analytical expressions [16]). Let us write down explicitexpressions for a couple of leading heat kernel coefficients. In the NC case wehave a ( P + m ) planarsub = 1(4 π ) / Z d x g ϕ ,a ( P + m ) planarsub = 1(4 π ) / Z d x (cid:20) g ϕ + g m ϕ (cid:21) . (89)The coefficients appearing in the commutative case are a ( P (0) + m ) sub = − π ) / Z d x g ϕ ,a ( P (0) + m ) sub = 1(4 π ) / Z d x (cid:20) g ϕ + g m ϕ (cid:21) . (90)In both cases the corresponding heat kernel coefficients differ only by numericalprefactors in front of the same powers of ϕ .The high temperature expansion does not depend on θ . In the limit θ → β → θ → β ≪ C | θ | which was imposed when studying the mixedcontributions to the asymptotic expansion.In space-space NC theories a drastic reduction of the degrees of freedom inthe non-planar sector above certain temperature was observed in [3]. This maybe related in some way to the absence of non-planar contributions to the hightemperature power law asymptotics in space-time NC theories found above.One can calculate the high-temperature asymptotics also in the real-timeformalism. The key observation that the non-planar sector does not contributeremains valid also in this formalism. The heat kernel expansion in the planarsector has the standard form (though the values of the heat kernel coefficientsdiffer from the commutative case). Therefore, one can repeat step by step thecalculations of [27] and obtain an expansion for the free energy which is consistentwith the expansion for the effective action derived above in the imaginary-timeformalism. This is another argument in favor of the assumptions made in sec.3.2. In this paper we considered some basic features of the finite-temperature NC φ theory in the imaginary-time formalism. We restricted ourselves to the caseof pure space-time noncommutativity, θ ij = 0. We used the zeta function reg-ularization and the heat kernel methods. Although we found highly non-localnon-planar heat kernel coefficients, such coefficients do not contribute neither tothe one-loop divergences, nor to the high temperature asymptotics. The theorycan be renormalized at one loop by making charge and mass renormalizations,as usual. The counterterms do not depend on the temperature (as long as it isnon-zero). We expect that the renormalization of this theory at zero temperatureproceeds differently. The high temperature expansion of the one-loop effectiveaction looks similar to the commutative case. The coefficients of this expansiondo not depend on the NC parameter θ , but again, one has to assume that thisparameter is non-zero.We have also studied relations between the imaginary and real time formu-lations. We found that the Wick rotation of the Euclidean free energy gives thecanonical free energy modulo two assumptions. One assumptions about the be-havior of the spectral density on the complex plane is of technical nature. Anotherone is more fundamental, it concerns the interpretation of the eigenfrequneciesof perturbations as one-particle energies.An extension of our results to more general models containing gauge fieldsand spinors can be done rather straightforwardly. Gauge fields are particularlyimportant to make connections to other approaches [6, 7]. Curved space-timeswill probably be difficult because of the problems with the heat kernel expansion.21ven in the case of a two-dimensional NC space with a non-trivial metric theheat kernel coefficients for a (rather simple) operator are known as power seriesin the conformal factor only [35]. Acknowledgements
One of the authors (D.V.V.) is grateful to C. Dehne for helpful discussions onnoncommutative theories and to D. Fursaev for answering endless questions re-garding the methods of non-linear spectral problem. This work was supported inpart by FAPESP and by the grant RNP 2.1.1.1112.
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