Thermal Field Theory and Generalized Light Front Quantization H

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2003 Thermal field theory and generalized light front quantization H. Arthur Weldon

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This Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarship by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. arXiv:hep-ph/0302147v1 17 Feb 2003 h eainof relation The r oeo h ih rn n rpsda h appro- the as proposed and front function light partition the- the priate field on thermal done and mechanics ory statistical might to coordinates over front carry light in quantizing of advantages T e htcmsfo h aihn fteo-hl energy, on-shell as the (1.1a), of prob- vanishing immediate Eq. the an from found comes they that theory lem field free a For [8]. h operator The aitna nta tgnrtsteeouino the of evolution the coordinate generates the it in fields that in Hamiltonian as h asselcniinepesdi em fthe of terms in expressed variables condition momentum shell mass the cause 6]. 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For exam- 0 0 3 √ α ple, the light front choice is expressed x = (x +x )/ 2. u nα > 0. (1.4c) µ It is also convenient that for Aα the superscript µ is nei- ther a first index nor a second index since µ cannot be The conventional rest frame choice is n0 = u0 = 1 and confused with α. nj = uj = 0. A Lorentz boost from the rest frame still It is misleading to refer to the general coordinate trans- gives nα = uα. The equality of these two vectors is not formations as a change in the reference frame. A change general as various examples will show. One can see that in the physical reference frame can only be done by ro- light front quantization using the partition function in tations and Lorentz boosts. Eq. (1.1c) fails, even apart from the divergence issues µ The 16 real numbers Aα must have a nonzero deter- mentioned previously, because it requires v 1 in order 0 α → minant. To guarantee that x be a time coordinate, it that Pαu be proportional to P . This makes TLC = T √1 v2/√2 0. In contrast, the− successful approach is necessary to require that the covariant Lorentz vector − → 0 of Alves, Das, and Perez [8] in Eq. (1.2) fits into the Aα be time-like or light-like: α α general framework above with n = (1, 0, 0, 1) and u = A0 A0 gαβ 0, (2.2) (1, 0, 0, 0). α β ≥ The purpose of this paper is to investigate thermal where gαβ = diag(1, 1, 1, 1) is the Minkowski met- − − − field theory formulated in general coordinates that are ric. It is sometimes useful to express the transformation arbitrary linear combinations of the Cartesian t,x,y,z matrix in terms of partial derivatives: and see if there are any computational advantages to µ µ ∂x other formulations. The method is conservative in that A = . (2.3) α ∂xα the new coordinates are restricted to be linear combinations of the Cartesian coordinates [9], a restriction which The relation (2.1) can be inverted to find the Cartesian coordinates xβ as linear combinations of the new coordi- guarantees that physical consequences will be the same ν for the following reason. The Lagrangians of fundamen- nates x . These partial derivatives are denoted tal field theories are invariant under arbitrary nonlinear ∂xβ Aβ = . (2.4) coordinate transformations in accord with the principle ν ∂xν of equivalence and thus are trivially invariant under the The transformations satisfy linear coordinate transformations considered here. Phys- µ α µ µ β β ically interesting quantities in thermal field theories are Aα Aν = δν Aα Aµ = δα. (2.5) either scalars, spinors, or tensors. Whether calculated The differential length element in Cartesian coordi- in Cartesian coordinates or more general linear coordi- α β 2 2 2 2 nates is gαβ dx dx = (dt) (dx) (dy) (dz) . In nates, the scalar quantities should be identical and the the new coordinates there is a− new metric− g − given by spinor and tensor quantities should be simply related by µν α β the chosen transformations. gµν = Aµ Aν gαβ, (2.6) Section II develops the formalism of thermal field the- α β µ ν ory in the general coordinates in order to determine the which satisfies gαβ dx dx = gµν dx dx . If all 12 of the possible choices for the surface of quantization and the off-diagonal entries in gµν vanish, the new coordinates velocity of the heat bath. Section III treats several exam- are orthogonal. This occurs, for example, if the A are ples that are specifically related to light front quantiza- Lorentz transformations. In general the gµν are not di- tion and it may be read independently of Sec. II. Section agonal and in general the new coordinates are oblique. IV provides some conclusions. It will be convenient for later purposes to denote the determinant of the covariant metric by

g = Det gµν < 0. (2.7) II. THERMAL FIELD THEORY IN OBLIQUE h i COORDINATES The Jacobian of the coordinate transformation dx0dx1dx2dx3 = Jdx0dx1dx2dx3, A. Transformed coordinates and metric is therefore From the Cartesian coordinates x0 = t, x1, x2, x3 a α J = Det Aµ = √ g. (2.8) new set of coordinates x0, x1, x2, x3 can be formed by h i − taking linear combinations: It will also be necessary to use the contravariant metric gµν , related to the Minkowski metric by µ µ α x = Aα x , (2.1) µν µ ν αβ g = Aα Aβ g . (2.9) µ where the Aα are a set of 16 real constants. The notation The requirement in Eq. (2.2) can be stated as used is due to Schouten [10]. It expresses the fact that the g00 0. (2.10) space-time point speciﬁed by the four-vector x does not ≥ 3

B. Four-momentum operators coordinates xα xα + δxα with δxα = Aα δx0. Using → 0 α 0 Pα δx = P0 δx the shifted field operator is In the conventional quantization at fixed x0 the mo- α α mentum operators P are the generators of space-time φ(x + δx) = exp(iPα δx )φ(x)exp( iPα δx ) λ − translations. For a scalar field operator φ(x) they satisfy = exp(iP δx0)φ(x)exp( iP δx0). 0 − 0 ∂φ P , φ = i . For evolution in imaginary values of δx0 to be equiva- λ − ∂xλ lent to thermal averaging requires that the density oper- The linear combination of these generators defined by ator involve only P0 and not Pj . Therefore the spatial components of the contravariant velocity must vanish: λ 1 2 3 Pµ = Aµ Pλ, (2.11) u = u = u = 0. This describes a heat bath that is at rest in the oblique coordinates. The normalized velocity λ with Aµ given in Eq. (2.4) will satisfy vector is 1 ∂xλ ∂φ ∂φ uσ = , 0, 0, 0 . (2.15) Pµ, φ = i = i . (2.12) g − ∂xµ ∂xλ − ∂xµ √ 00 Note that this imposes a new requirement on the metric, In particular, P0 generates the evolution in the variable x0. Appendix B shows that if one quantizes at fixed x0 then the Hamiltonian is this same operator P . 0 g > 0, (2.16) Note that the temporal evolution is in the variable 00 0 0 α x = Aα x but the evolution operator is in a differ- that is different than Eq. (2.10). This condition prevents ent linear combination: P = Aα P . These two vectors 0 0 α standard light front quantization since g00 would vanish. 0 α In all subsequent discussion it will be assumed that Eq. are reciprocal in the sense that Aα A0 = 1. Lorentz transformations are special in that the metrics are the (2.16) is satisfied. The density operator is same, g = gαβ and the two vectors are the same, αβ ̺ˆ = exp βP √g , (2.17) A0 = g Aβ. − 0 00 α αβ 0 or equivalently Eq. (2.13) if the velocity vector is expressed in Cartesian coordinates: C. Density operator α α µ α u = Aµ u = A0 √g00 . (2.18) Field theory at finite temperature requires a density 0 operator and it is not obvious exactly what the density The vector normal to the quantization surface is nα = Aα α operator should be when using general oblique coordi- and therefore nαu =1/√g00 > 0 as stated in Eq. (1.4c). nates. For conventional quantization at fixed x0 the den- The density operator is used to define the thermal av- sity operator has the form eraged Wightman function:

̺ˆ = exp βP uα , (2.13) >(x) = Tr exp βP0 /√g00 φ(x)φ(0) Z, (2.19) − α D − where uα is some four-velocity. Rewriting this in general where Z = Tr[ˆ̺] is the partition function. Under imagi- 0 σ nary shifts in the oblique time of the form δx = iα/√g00, coordinates gives̺ ˆ = exp βPσ u . The appropriate n − o the behavior is value of uσ is undetermined but since the Minkowski four- α β 0 j velocity satisﬁes u u g = 1, the velocity in oblique > x + iα √g , x (2.20) αβ D 00 coordinates must satisfy (β + α)P αP = Tr exp 0 φ(x)exp 0 φ(0) Z. σ µ − √g √g u u gσµ =1. (2.14) 00 00

The spectrum of P0 is bounded from below but will always have arbitrarily large positive eigenvalues. If the 1. KMS relation two exponents are negative then the inﬁnitely large energies will be suppressed. In other words, (2.20) is analytic With a partition function of the above form, the ther- for β α 0. For the choice α = β or equivalently mal Wightman function is δxα−= ≤iβuα≤the result is − − >(x) = Tr ̺ˆ φ(x)φ(0) Tr[ˆ̺ ]. 0 j 0 j > x iβ √g , x = > x , x . (2.21) D D − 00 D − − Consider a shift the oblique time, i.e. x0 x0 + δx0 This is the Kubo-Martin-Schwinger relation [11] ex- → with ﬁxed values of x1, x2, x3. In terms of the Cartesian pressed in oblique coordinates. 4

2. Tolman’s law operator in covariant form:

α λ The dependence of the partition function (2.17) on the ̺ˆ = exp β dSαT λu . (2.24) n − Z · o combination P0 T√g00 merits further discussion. One explanation comes from perturbation theory in which at each order the eigenvalues of the operator P0 are the sum D. Thermal field theory in real time of single particle energies p0, each satisfying a mass shell µν 2 condition g pµpν = m , for various masses. Among the 0 allowed coordinate transformations are scale transforma- To quantize a field theory at a fixed value of x is tions. Rescaling the contravariant time by a factor λ, straightforward but there are some unfamiliar aspects that originate from the oblique metric g being non- as in x0 λ x0, would rescale the covariant energy by µν diagonal. Appendix B performs the explicit quantization p p /λ→. The combination P /T would not be invari- 0 0 0 for a scalar field theory. It is most natural to deal with ant→ under this transformation. However, the scale trans- contravariant space-time coordinates xµ and covariant formation changes the covariant metric, g g /λ2, 00 00 momentum variables p so that the solutions of the field and the combination P T g is invariant. → µ 0 √ 00 equations are superpositions of plane waves of the form There is another way to understand why the partition exp( ip xµ). For spinor particles the Dirac matrices are function (2.17) depends on the combination T g . A µ √ 00 γµ =±Aµγα and satisfy γµ,γν =2gµν. For gauge boson condition necessary for thermal equilibrium in inertial α propagators there are natural{ generalizations} of Coulomb coordinates is that the temperature should be uniform gauge, axial gauge, and covariant gauges. This section in space and time. Tolman [12] investigated the condi- will summarize some familiar results but expressed in tions for thermal equilibrium in a gravitational field and oblique coordinates. Either canonical quantization or showed that a temperature gradient is necessary to pre- functional integration may be used [16, 17]. vent the flow of heat from regions of higher gravitational potential to regions of lower gravitational potential. The quantitative result is summarized by the statement that 1. Propagators the product T√g00 must be constant and is known at Tolman’s law [13]. An alternative derivation is given by Landau and Lif- At zero temperature the free propagator for a scalar shitz [14], who discuss how to compute the entropy in field is the microcanonical ensemble in a general curvilinear co- 1 D(p0,pj )= µν 2 . (2.25) ordinate system (with or without gravity). In the mi- g pµpν m + iǫ crocanonical ensemble, the temperature is computed by − differentiating the entropy with respect to the energy and The integration measure over loop momenta is this leads to the result that T√g00 must be constant. dp dp dp dp 0 1 2 3 . (2.26) Z √ g(2π)4 − 3. Covariant density operator At non-zero temperature the propagator has the usual 2 2 matrix structure [16, 17]. The Bose-Einstein or For later purposes it is convenient to express the den- Fermi-Dirac× functions become sity operator (2.17) in a covariant form in terms of the ν 1 conserved energy-momentum operator T µ. The gener- n = . (2.27) · exp(β p /√g ) 1 ators of translations are | 0| 00 ∓ 1 2 3 0 As discussed later, the most interesting possibility is Pµ = √ g dx dx dx T µ. 00 − Z · to choose a coordinate transformation such that g = 0. This makes the denominator of the propagator linear in Rewrite this in terms of the energy-momentum operator α p0 and therefore there is only one pole in (2.25). in Cartesian coordinates, T λ as · If g00 = 0 then the propagator has poles at two val- 6 α λ ues of p0 but the positive and negative values of p0 will Pµ = dSα T λ Aµ, (2.22) Z · have different magnitudes. It is sometimes convenient to express the propagator in a mixed form, in terms of the in accordance with Eq. (2.11). The three-dimensional, covariant energy p but the contravariant momenta pj . differential surface element is 0 To do this, use the identity 0 1 2 3 dSα = √ g Aα dx dx dx . (2.23) − µ ν 1 0 i 2 goig0j i j gµν p p = g00 p + g0i p + gij p p This surface of quantization is orthogonal to three con- g00 − g00 α travariant: vectors A dSα = 0 for j = 1, 2, 3. Contract- 2 j (p0) i j µ = γij p p ing Pµ with u and using Eq. (2.15) gives the density g00 − 5

where γij is given by where ρ is the energy density. The ﬁrst law of thermodynamics guarantees that there is an additional state goig0j function, the entropy density σ, related to energy density γij = gij + . (2.28) − g00 and pressure by

The determinant is Det[γij ] = g/g00 > 0 [18]. Define T σ = ρ + P. (2.36) the effective energy − This relation is equivalent to the more familiar differen- 1/2 2 i j tial relation T dS = dU + P dV . E = m + γij p p . (2.29) h i It is convenient to express the right hand side of Eq. The same propagator in these variables is (2.33) as the trace of the covariant density operator (2.24). The left hand side of (2.33) can be written covari- j g00 antly in terms of the differential surface element (2.23) D(p0,p )= 2 2 (2.30) (p0) g00 E + iǫ 1 2 3 α − using √ g dx dx dx /√g00 = dSαu as an integral over the free− energy density Φ: The poles are now at p0 = √g00 E. The Bose-Einstein or Fermi-Dirac functions become± F/ g = dS uαΦ. 1 √ 00 α n = . (2.31) Z exp(βE) 1 ∓ The manifestly covariant statement of Eq. (2.33) is The integration over covariant energy and contravariant three-momenta is α α λ exp β dSαΦu = Tr exp β dSαT λu . − − · 1 2 3 n Z o h n Z oi √ g dp0 dp dp dp (2.37) − 4 . (2.32) g00 Z (2π) Now apply this to two different equilibrium states with infinitessimally different β and uα. The difference gives the differential relation 2. Statistical mechanics α α Tr ̺Tˆ λ λ d(βΦu )= · d(βu ). (2.38) The partition function provides a direct calculation of Tr[ˆ̺ ] the thermodynamic functions via the Helmholtz free energy F : The differential on the left hand side is d(βΦ)uα + (βΦ)duα. exp βF/√g00 = Tr exp βP0/√g00 . (2.33) n − o h − i Using the thermal average of the energy-momentum ten- The free energy F (T, V ) then allows computation of the sor (2.35), the right hand side is pressure and entropy: (ρ + P )uαu Pδα dβuλ + βduλ ∂F λ − λ P = (2.34a) = ρdβuα Pβdu α, −∂V T − λ ∂F where uλdu = 0 has been used. Equating the left and S = , (2.34b) α α − ∂T V right hand sides and noting that u and du are orthogonal vectors, gives the two relations and the energy is U = F + TS. Φ= P (2.39a) − 3. Covariant statistical mechanics d(βΦ) = ρdβ. (2.39b) The first relation is the same as Eq. (2.34a). The second The prescriptions in Eq. (2.34) can be derived rather relation implies dΦ = (Φ ρ)dT/T = (P + ρ)dT/T = elegantly from the partition function if it is formulated σdT . Therefore σ = −∂Φ/∂T , which− is the same as covariantly [13, 15]. In global thermal equilibrium con- −Eq. (2.34b) but expressed− in terms of densities. sidered here, the temperature T and the velocity uµ of the heat bath are independent of space-time and so there is no heat flux or viscosity. The thermal average of the E. Thermal field theory in imaginary time energy-momentum operator has the perfect fluid form:

νµ It is also interesting to quantize in imaginary oblique Tr ̺Tˆ 0 h i = (ρ + P )gνµ Puν uµ, (2.35) time by letting x = iτ with 0 τ β/√g00. As Tr ̺ˆ − shown in Eq. (2.20) the− thermal Wightman≤ ≤ function is 6 analytic for τ in this region. Since the Cartesian coordi- of a,b,c,d. If a > b and d > c then x0 is a true time | | | | | | | | nates are linear combinations of the oblique coordinates, coordinate and x3 is a true space coordinate. Light front α α µ 0 x = Aµ x , making x pure imaginary makes some of coordinates violate this, viz. a = b and c = d . The the Cartesian coordinates complex. In other words, going inverse transformation to (3.1a)| | is | | | | | | to imaginary time does not commute with the coordinate x0 = (dx0 bx3)/N transformations. − The Euclidean propagator comes from the replacement x3 = ( cx0 + ax3)/N, (3.1b) p iωn in Eq. (2.25) where ωn =2πnT √g : − 0 →− 00 where N = ad bc = 0. The two conditions − 6 1 d DE(p)= − . (2.40) a> 0, > 0, (3.1c) g00ω2 +2ig0jω p gij p p + m2 N n n j − i j guarantee that increasing values of x0 corresponds to in- Note that the denominator has an imaginary part be- 0 cause of the non-diagonal metric but the real part of the creasing x . denominator is positive as always. In perturbation the- (1) Metric. The contravariant metric is ory the summation/integration over loop momenta is a2 b2 0 0 ac bd −0 10− 0 gµ ν =  −  , (3.2a) T √g00 ∞ dp dp dp 0 0 1 0 1 2 3 . − √ g Z (2π)3  ac bd 0 0 c2 d2  n=X   − −∞ − − and the covariant metric is One can canonically quantize in imaginary time, in d2 c2 0 0 ac bd which case the field operators obey equations of motion. −0 N 2 0− 0 1 Alternatively one can use a a Euclidean functional in- g =   . (3.2b) µ ν 0− 0 N 2 0 2 tegral [16, 17, 19] in which case the fields are periodic N  ac bd 0− 0 b2 a2  under τ τ + β/√g00. The partition function is  − −  → The necessary requirement g = 0 implies d = c ; how- 00 6 | | 6 | | 4 ever, there is nothing wrong with choosing a = b . The Z = [Dφ]exp d xE , (2.41) Z Z L 3 3 matrix defined in Eq. (2.28) is | | | | periodic n o × where the integration element in four-dimensional Eu- 10 0 clidean space-time is given by γij =  01 0  . (3.3) 2 2 1 0 0 (d c )− β/√g00  −  4 1 2 3 d xE = √ g dτ dx dx dx . (2) Momenta. The contravariant form of the oblique Z − Z Z ν ν α 0 momenta are p = Aα p in parallel with Eq. (3.1a). From this, the covariant momenta are obtained by ap- Thermodynamics is computed from the Helmholtz free plying the metric, p = g pν , with the result energy F (T, V ) µ µν 0 3 p0 = (dp + cp )/N 0 3 exp βF/√g00 = Z (2.42) p = (bp + ap )/N, (3.4) n− o 3 − 1 2 using Eq. (2.34). and, of course, p1 = p , p2 = p . (3) Density operator.− The time− evolution operator is α ∂x dP0 cP3 P = Pα = − , III. EXAMPLES 0 ∂x0 N and therefore the density operator is It is easy to see that time-independent transformations 0 0 d P0 cP3ǫ(d) will not give anything new. More specifically, if x = x exp βP0 /√g0 0 = exp β | | − , 0 1 2 3 − − √d2 c2 then the quantization is at fixed x and if x , x , and x − are independent of x0 then Aj = 0 so that uj = 0 and after using Eq. (3.1c). Here ǫ(d) = 1 is the sign func- 0 ± the density operator will be exp βP . tion. As expected, d = c is excluded. One can under- 0 | | | | This section will deal with a general{− } class of examples stand the form of the density operator in a more physical based on transformations of the form way. Since the heat bath is at rest in the x3 coordinate, its laboratory velocity is v = dx3/dx0 = c/d from Eq. 0 0 3 2 1/2 − x = ax + bx (3.1a). Using γ = (1 v )− gives the four-velocity 3 0 3 − x = cx + dx , (3.1a) d cǫ(d) U α = γ, 0, 0, γv = | | , 0, 0, − (3.5) 1 1 2 2 √d2 c2 √d2 c2 with x = x and x = x always understood. All the − − examples in this section will result from special choices Thus the density operator is̺ ˆ = exp( βP U α). − α 7

A. Lorentz transformations and generalizations (2) Choice of Alves, Das, & Perez. The calculations in [8] can be stated as the choice a = b = d =1,c = 0: (1) Quantization at rest but moving heat bath. The x0 = x0 + x3 most intuitive situation physically is to quantize con- 3 3 ventionally at ﬁxed x0 but for the heat bath to have x = x . (3.12) a velocity v. This is easily accomplished by the choice The density operator becomes exp βP . The covari- a =1,b =0,c = γv,d = γ so that 0 ant metric is {− } x0 = x0 1 0 0 1 0 1 0− 0 x3 = γ(x3 + vx0). (3.6) g =   , (3.13) µ ν 0− 0 1 0  1 0− 0 0  The density operator is exp βγ(P0 vP3) .  −  − − (2) Quantization surface and heat bath moving diﬀer- ν 0 and so the covariant coordinates are x0 = g0ν x = x and ently. A more general option is to give the quantization ν 0 3 + x3 = g3ν x = (x + x )= √2x . The corresponding surface a velocity v′ and the heat bath a velocity v: covariant momentum− components− are

0 0 3 0 x = γ′(x + v′x ) p0 = p 3 3 0 p = (p0 + p3)= √2p+. x = γ(x + vx ) (3.7) 3 − − The contravariant metric is However, only the velocity v enters into the density operator:̺ ˆ = exp βγ(P0 vP3) . Note that the metric 0 0 0 1 − − 0 1 0− 0 will depend on v and v′. gµ ν =   . (3.14) 0− 0 1 0  1 0− 0 1   − −  B. Light front and generalizations Because g00 vanishes, the momentum space propagator is linear in p0: As stated previously, strict light front coordinates in µν 2 2 2 which d = c are forbidden. g pµpν = 2p0p3 (p1) (p2) (p3) | | | | − − − − (1) Front moving at v< 1. An interesting case is the = 2√2p0p+ (p1)2 (p2)2 2(p+)2. transformation − − − In the Euclidean formulation the contravariant time be- x0 = x0 cos(θ/2)+ x3 sin(θ/2) comes negative, imaginary: x0 iτ; the covariant energy becomes discrete, imaginary:→ −p iω with x3 = x0 sin(θ/2) x3 cos(θ/2), (3.8) 0 → − n − ωn = 2πnT . (Note g00 = 1.) The Euclidean propagator used in [8] is where π/2 < θ < π/2. These would be light front coordinates− if θ were allowed to take the value π/2. 1 ± . (3.15) The covariant and contravariant metrics are equal: 2i√2 ω p+ (p1)2 (p2)2 2(p+)2 m2 n − − − − cos θ 0 0 sin θ (3) ADP with moving heat bath. It is simple to modify 0 10 0 the previous case to allow for a moving heat bath. Choose g =   = gµν . (3.9) µ ν 0− 0 1 0 a = b = 1, c = γv, and d = γ so that  sin θ 0− 0 cos θ   −  x0 = x0 + x3 The density operator is x3 = γ(vx0 + x3). (3.16)

The density operator is exp βγ(P0 vP3) , corre- P0 cos(θ/2)+ P3 sin(θ/2) − − ̺ˆ = exp β , (3.10) 00 − cos θ sponding to a moving heat bath. As before g = 0 but now g = √1+ v/√1 v. 00 − and obviously fails at θ = π/2. Alternatively, one can use v = tan(θ/2) and express the transformation as IV. CONCLUSIONS

x0 = (x0 + vx3)/ 1+ v2 In standard light front quantization g = g = 0 p 00 ++ x3 = (vx0 x3)/ 1+ v2, (3.11) and this makes it impossible to formulate statistical me- − p chanics and thermal ﬁeld theory. Physically, the problem so that the density operator is exp βγ(P + vP ) . is the inﬁnite velocity of the light front. − 0 3 8

The most interesting possibility is to choose oblique field and then calculate the thermal average of the free 00 coordinates which satisfy g00 = 0 but g = 0 as in Eq. energy-momentum tensor. (3.12), which is the case studied6 by Alves, Das, and Perez [8]. The advantage of choosing g00 = 0 is that the denominator of the propagator, gµν p p m2, will be linear in 1. Equation of motion µ ν − the energy variable p0. Consequently the propagator will have only one pole and not two. This reduces the com- The action expressed as an integral over contravariant putational effort required for multiloop diagrams. Any coordinates is √ g dx0dx1dx2dx3 with Lagrangian diagram for which the kinematics allows N propagators − L density R to be on shell would normally produce 2N contributions. However if g00 = 0, there will be only one contribution. 1 ∂φ ∂φ 1 = gµν m2φ2. (B1) A very straightforward application would be to com- L 2 ∂xµ ∂xν − 2 pute the quark and gluon propagators in the hard thermal loop approximation using Eq. (3.12) and verify the The field equation that follows from the Lagrangian den- rotational invariance of the dispersion relations [17]. A sity is more ambitious task would be to compute the vertex functions in the hard thermal loop approximation. ∂2φ gµν = m2φ. (B2) ∂xµ∂xν Acknowledgments The solution to this will be a superposition of plane waves of the form exp ip xα , with the phase expressed in − α It is a pleasure to thank Ashok Das for sending me his terms of contravariant spatial coordinates and covariant paper [8] and thereby arousing my interest in the subject. momentum coordinates. The equation of motion gives This work was supported in part by a grant from the U.S. the mass shell condition National Science Foundation under grant PHY-0099380. µν 2 g pµpν = m . (B3)

APPENDIX A: LORENTZ INVARIANCE OF gµν This is a quadratic equation for p0 with two solutions

0j 2 ij 1/2 The general non-diagonal metric gµν is always invari- g pj m + pipj Γ ant under three rotations and three Lorentz boosts. How- p0 = (B4) ± − g0 0 ± g00 ever the representation of these six transformations depend on the coordinate system xµ. where Γij is the 3 3 matrix The usual representation of a Lorentz transformation′ × α from′ one set of Cartesian coordinates to another, x = 0i 0j α α ij g g ij Λα x , leaves invariant the Minkowski metric tensor: Γ = g . (B5) 00 − ′ ′ g α β ′ ′ Λα Λβ gα β = gαβ. (A1) 0 j The two solutions are exp( ip0 x ipjx ). The ener- As before, the index notation of Schouten [10] is used. − ± − gies p0 are not invariant under momentum inversion, Here α′ runs over 0′, 1′, 2′, 3′. The Minkowski metric is ± but rather p0 p0+ when pj changes sign. Therefore invariant: g ′ ′ = g = 1, g ′ ′ = g = 1, etc. − →− 0 0 00 1 1 11 − one can use for the second plane wave the negative mo- Each Lorentz transformation of the Cartesian coordi- mentum solution exp(ip x0 + ip xj ). The solution to nates induces a Lorentz transformation of the oblique 0+ j µ µ ρ the ﬁeld equation can be expanded as coordinates, x = Wρ x , where

′ µ µ α α dp1dp2dp3 ip x ip x W = A ′ Λ A (A2) φ(x)= a(p)e− · +a(p)†e · . (B6) ρ α α ρ Z √ g(2π)32 p0 Because Λ keeps the Cartesian metric invariant, W au- − | | tomatically keeps the oblique metric invariant: where p x = p x0 + p xj . The contravariant energies · 0+ j µ 0 ν p ± are equal in magnitude: Wρ Wσ gµν = gρσ . (A3)

0 00 0j p ± = g p0 + g pj APPENDIX B: QUANTIZATION IN OBLIQUE ± 1/2 COORDINATES = g00 m2 + p p Γij , (B7) ±q i j This section will perform the explicit quantization in an arbitrary oblique coordinate system for the free scalar and p0 will be denoted simply by p0 [18]. | | 9

2. Canonical quantization The Hamiltonian requires integrating over the contravariant three-volume For quantization on the surfaces of constant x0, the 1 2 3 canonical momentum is P0 = √ g dx dx dx . − Z H ∂ ∂φ ∂φ π(x)= L = g0µ = (B8) Working this out explicitly gives 0 µ ∂ ∂φ/∂x ∂x ∂x0 dp1dp2dp3 p0 The explicit mode expansion is P0 = a†(p)a(p)+ a(p)a†(p) . Z √ g(2π)32p0 2 h i − (B11) dp1dp2dp3 ip x ip x π(x)= i a(p)e− · a(p)†e · Note that the covariant energy p in the numerator does − Z √ g(2π)32 − 0 − not cancel the contravariant energy p0 in the denomina- If the mode operators are required to satisfy tor. The commutation relation

3 ∂φ 0 3 P0, φ(x) = i a(p),a†(p′) = √ g2p (2π) δ(p p′ ), (B9) − ∂x0 − j − j jY=1 veriﬁes that the Hamiltonian is the generator of transla- then equal time commutator has the correct value: tions in the contravariant time variable x0.

π(x), φ(x′) x0=x′0 5. Energy and momentum dp dp dp j ′j j ′j 1 2 3 ipj (x x ) ipj (x x ) = i 3 e− − + e − − Z (2π) 2 h i The canonical energy-momentum tensor is i 1 1 2 2 3 3 = δ(x x′ )δ(x x′ )δ(x x′ ). −√ g − − − − ν ∂φ ∂φ ν T µ = µ δµ (B12) · ∂xν ∂x − L

3. Microcausality ν ν and satisfies the conservation laws ∂T µ/∂x = 0. The · µ = 0 and µ = m components of this equation are It is easy to verify microcausality. The commutator of two fields is 0 n ∂T 0 ∂T 0 0= · + · (B13) α α 0 n dp1dp2dp3 ipαx ipαx ∂x ∂x φ(x), φ(0) = e− e . Z 3 0 − 0 n √ g(2π) 2p h i ∂T m ∂T m − 0= · + · (B14) 0 ∂xn Change to Minkowski integration variables by defining ∂x λ λ α λ pα = pλ∂x /∂x so that pαx = pλx . The integration 0 0 From Eq. (B10), = T 0 and thus T 0 is the energy measure is invariant and therefore H · · n density. The first of the above equations indentifies T 0 0 · dp1dp2dp3 ip x ip x as the energy flux. From the second, T is the momen- φ(x), φ(0) = e− · e · . m Z (2π)32p0 − tum density and T n is the momentum· flux. Integrating h i m the energy and momentum· densities over a contravariant This is the conventional answer for the commutator. It three-volume gives λ vanishes for space-like separations xλx < 0. Since α λ α xαx = xλx it vanishes for xαx < 0. 1 2 3 0 Pµ = √ g dx dx dx T µ. (B15) − Z · 4. Hamiltonian These integrals are independent of the contravariant 0 time: ∂Pµ/∂x = 0. They generate translations in the The canonical Hamiltonian density is contravariant coordinates: ∂φ ∂φ = π (B10) Pµ, φ(x) = i µ . (B16) 0 − ∂x H ∂x − L It is convenient to express this in terms of mixed con- The explicit form for the three-momentum operators is travariant and covariant derivatives: dp1dp2dp3 pj P = a†(p)a(p)+ a(p)a†(p) . 1 ∂φ ∂φ ∂φ ∂φ ∂φ ∂φ ∂φ ∂φ j Z √ g(2π)32p0 2 = +m2φ2 h i 0 1 2 3 − H 2∂x0 ∂x − ∂x1 ∂x − ∂x2 ∂x − ∂x3 ∂x (B17) 10

6. Thermal Averages whose evaluation is standard:

This section will show that despite the somewhat com- plicated dispersion relations in Eqs. (B4) and (B7), the α β Tµν = AµAν (ρ + P )uαuβ Pgαβ thermal average of the energy-momentum tensor can be h i − computed directly and gives the conventional answer. = (ρ + P )uµuν Pgµν . (B21) − Bose-Einstein statistics gives for the thermal average of the energy-momentum tensor of a free gas of scalar particles The ﬁnal result is expressed in terms of the oblique velocity vector and the oblique metric tensor. The most dp1dp2dp3 pµpν Tµν = . (B18) physical quantity is the mixed tensor h i Z √ g(2π)3p0 exp βp /√g 1 − 0 00 − To perform this integration, change to Minkowski momenta kα, where ρ 0 0 0 µ  (ρ+P )g01/g00 P 0 0  α T ν = − , (B22) pµ = Aµ kα. (B19) h · i (ρ+P )g /g 0 P 0 02 00 −  (ρ+P )g03/g00 0 0 P  2 2 1/2  −  The mass shell condition requires k0 = (k +m ) . And λ λ λ 0 p0/√g00 = kλu with u = A0 u , the oblique velocity given by Eq. (2.15). The change of variables gives where Eq. (2.15) has been used. The oﬀ-diagonal entries 0 in the metric give a nonzero momentum density: T n = α β dk1dk2dk3 kαkβ h · i Tµν = A A , (B20) (ρ + P )g /g . h i µ ν Z (2π)3k exp(βk uλ) 1 0n 00 0 λ −

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