The relationship between field theory in the simplest space-times with a nontrivial topology and field theory at a finite temperature

Volume 82A, number 5

PHYSICS LETFERS

30 March 1981

THE RELATIONSHIP BETWEEN FIELD THEORY IN THE SIMPLEST SPACE—TIMES WITH A NONTRIVIAL TOPOLOGY AND FIELD THEORY AT A FINITE TEMPERATURE N.I. KOCHELEV High-Energy Physics Institute, Academy of Sciences of the Kazakh SSR, Alma-Ata, USSR Received 22 January 1981

3 and It is shown that the main results of papers dealing with quantum field theory in space—time with topology S’ x R in flat space—time with parallel conducting plates can be obtained from field theory at a finite temperature. The new consequences of this fact axe discussed.

Recently Toms [1] pointed out the analogy between field theory in space—time with topology S1 X R3 (periodic identification in one of the spatial coordinates) and field theory at a finite temperature. In this note we draw attention to the close relationship between field theory at a finite temperature and field theory in space—time with topology S1 X R3 and to that in flat space—time with parallel conducting plates (this topology will further be referred to as K) at zero temperature. In fact, the Feynman rules for finite temperature are derived by means of a transition to the euclidean metric with the identification period equal to lIT over the euclidean time axis. Then the Green’s functions of fields are determined by the boundary conditions giving eigen-frequency spectrum the system. E.g., the the eigen-frequency spectrum of aofscalar field at a temperature Tis Wn = 2irnT, where n = 0, ±1, ±2 For the same field in S1 X R3 and K space—times the boundary conditions give the spectra = 2irn/L and w,~= irn/L, respectively. In the first case L is an identification period, while in the second one L is the distance between the plates. All the coordinates in a euclidean metric have equal rights, Hence, the Feynman rules in S1 X R3 and K space— times at zero temperature almost coincide with those for fields at a finite temperature. The only difference is that the role of temperature is played by the quantity I/L in S’ X R3, and by 1/2k in K. Thus, in order to determine the meaning of some quantity calculated

by a diagrammatic technique in S’ X R3 and K space— times it is enough to produce a simple substitution of variables in the expression for the same quantity in quantum field theory at finite temperature. This fact allows one to establish a relationship between important field characteristics in the two theories. To illustrate this let us take some examples. It is easy to show that the thermodynamical potential density of a free massive scalarfield, &~Z(T),defined by the expression m2T2 K 2(sm/T) ~2(T) = 2 2 2ir s1 s where m is the field mass and K2 is the Bessel function1 [2], transforms to the magnitude this quantity X R3 and K space—times by the of substitutions T in S -÷ l/L and T-÷ 1/2L, respectively. It should be noted that after the substitution T—~l/2L we have reproduced the result for the vacuum energy density of a massive scalar field in the K space—time obtained earlier in ref. [3] on other grounds. For the second example, we will consider the effective potential for the X~4/4!theory, because its temperature dependence, as was noted in ref. [4], gives valuable information on the presence of spontaneous symmetry breaking at finite temperature. In this theory the dynamical mass, up to the first order in A, depends on temperature as follows: m2 = AT2! 24 [4]. Thus, m2 = X/24L2 for S1 X R3 space—time and m2 = A/96L2 for K space—time, and it coincides with the —

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Volume 82A, number 5

PHYSICS LETTERS

results [5] for the so-called topological mass. Moreover, the basic conclusions of ref. [6] on spontaneous symmetry breaking in S1 X R3 for the A~4/4!theory and of ref. [7] where scalar electrodynamics has been considered in the same space—time, followin an elementary way from the results of ref. [4] by the substitution T—~i/L. So, important field characteristicsin the S1 X R3 and K space—times can be easily derived from the results already known in field theory at finite temperature. Now let us consider some consequences from the correspondence revealed for quantum chromodynamics in s1 X R3 and K space—times. The vacuum energy density up to the third order in the gauge coupling constant g for these space—times follows directly from the results of ref. [8]. For S1 X R3: 45L4

~ —

60L4

36L4

[(~N~+ ~)3I2+ ~2]

,

where Nf is the number of quark flavours, and for the K space—time an additional factor l/2~appears. If one uses the temperature dependence of the charmed pair production probability per unit time in a unit volume of plasma consisting of u, d and s quarks as obtained in ref. [9]: dW~~/d4x = (103a~/181r2)mcT3exp(—2m~/T), ~‘

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where a5 = g2/4ir, m~is the mass of the charmed quark, then it is evident that dW~~/d4x = (l03a~m~/ 18ir2L3) exp(—2m~L), m~L~ 1, in S1 X R3 space—time. Due to nontrivial topology this effect of pair production is analogous to evaporation of a static black hole (the Hawking effect). As a conclusion, it should be noted that the afore-cited examples do not cover all the consequences of the relations stated here. The author is grateful to E.V. Shuryak and V.1. Rus’kin for useful discussions and to Zh.S. Takibaev for his interest in this work and support. References

l8V2irL4

m~ T,

30 March 1981

[1] D.J. Toms, Phys. Rev. D21 (1980) 928. [2] L.D. Landau and S.Z. Belen’ky, Usp. Fiz. Nauk 56

(1955)

309. [3] N.I. Kochelev, Fix. Phys. 33 (1981) [4] L. Dolan and R.Yad. Jackiw, Rev. 571. D9 (1974) 3320. [5] D.J. Toms, Phys. Rev. D21 (1980) 2805. [6] G. Denardo and E. Spalluci, Nuci. Phys. B169 (1980)

514.

[7] D.J. Toms, Phys. Lett.

77A (1980) 303. [8] M. Kiguchi, Prog. Theor. Phys. 63 (1980) 146. [9] E.V. Shuryak, Phys. Rep. 61(1980) 71.