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Thermal Field Theory and Generalized Light Front Quantization H Faculty Scholarship 2003 Thermal field theory and generalized light front quantization H. Arthur Weldon Follow this and additional works at: https://researchrepository.wvu.edu/faculty_publications Digital Commons Citation Weldon, H. Arthur, "Thermal field theory and generalized light front quantization" (2003). Faculty Scholarship. 395. https://researchrepository.wvu.edu/faculty_publications/395 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]. Thermal field theory and generalized light front quantization H. Arthur Weldon Department of Physics, West Virginia University, Morgantown, West Virginia 26506-6315 The dependence of thermal field theory on the surface of quantization and on the velocity of the heat bath is investigated by working in general coordinates that are arbitrary linear combinations of the Minkowski coordinates. In the general coordinates the metric tensor gµν is non-diagonal. The Kubo, Martin, Schwinger condition requires periodicity in thermal correlation functions when the temporal variable changes by an amount i ( T √g00 ). Light front quantization fails since g00 = 0, − however various related quantizations are possible. 0 + I. INTRODUCTION variables does work. Using P = (P + P −)/√2 the partition function is A. The light front and thermal field theory + Tr exp (P + P −)/√2 T , (1.2) − For many years light front quantization has been ap- h n oi plied to deep inelastic scattering and the Wilson opera- instead of Eq. (1.1c). They performed one-loop calcula- tor product expansion. More recently it has been used in tions of the self-energy in scalar field theories with either QCD to compute hadron structure [1, 2, 3, 4]. Light front a φ4 interaction or a φ3 interaction. The final results for quantization brings both conceptual and computational both calculations were exactly the same as the conven- simplifications to certain hadronic processes [5, 6]. tional answers. Much of the computational simplification occurs be- cause the mass shell condition expressed in terms of the 0 3 B. Thermal field theory in generalized coordinates momentum variables p± = (p p )/√2 is ± p2 + m2 In order to explore the various possible options it is p− = ⊥ . (1.1a) 2p+ most efficient to consider quantization in a general set of space-time coordinates and later examine light front 0 3 The operator P − = (P P )/√2 plays the role of the quantization as a special case. The metric signature is Hamiltonian in that it generates− the evolution of the (+, , , ). fields in the coordinate x+ = (x0 + x3)/√2 The− − conventional− approach is to impose quantization conditions on fields at a fixed value of x0. The opera- ∂φ tor that generates time evolution is P . The partition P −, φ(x) = i . (1.1b) 0 − ∂x+ function is Recently Brodsky suggested [7] that the computational Tr exp P0/T , (1.3) advantages of quantizing in light front coordinates might − carry over to statistical mechanics and thermal field the- as is appropriate for a heat bath at rest. If this sys- ory done on the light front and proposed as the appro- tem is viewed from a Lorentz frame with velocity v and arXiv:hep-ph/0302147v1 17 Feb 2003 priate partition function four-velocity uα = (γ, 0, 0, γv), then the quantization α will be at a fixed value of x uα; the evolution oper- Tr exp P −/T . (1.1c) α − LC ator will be Pαu ; and the partition function will be α α Tr[exp( βPαu )]. The fact that u serves as both the The relation of TLC to the usual invariant temperature vector normal− to the surface of quantization and the T was unspecified. velocity vector of the heat bath is a unique feature of The suggestion was pursued by Alves, Das, and Perez Lorentz boosted coordinates and is not be true in more [8]. For a free field theory they found an immediate prob- general coordinates. lem that comes from the vanishing of the on-shell energy, The present paper investigates the dependence of ther- Eq. (1.1a), as p . The breakdown is not specific + → ∞ mal field theory on the surface of quantization and on the to the canonical ensemble. In the microcanonical ensem- velocity of the heat bath. Subsequent analysis will show ble the entropy is a measure of the multiparticle phase that for any vector nα that is time-like or light-like, it is space available to a gas of particles whose total energy is possible to quantize at a fixed value of n xα, fixed. To any configuration with a fixed value of the to- α tal P − one can add any number of zero-energy particles π(x), φ(0) δ(n xα c)= iδ4(x), (1.4a) + α − − each having an infinite value of p . The entropy of such a configuration diverges. and employ the partition function Alves, Das, and Perez [8] showed that a thermal av- erage performed in the rest frame but using light-front Tr exp P uα/T , (1.4b) − α 2 appropriate to a heat bath moving with four-velocity change under a coordinate transformation; only the la- α α u = (γ, 0, 0, γv), provided only that nα and u satisfy bels used to indentify the components change. 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 combina- tions 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 de- terminant 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 .
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