Why the real time formalism doesn’t factorise ∗

Why the real time formalism doesn't factorise * Abstract We show that in the real time formalism, the generating functional for thermal Green functions does not factorise. However for most calculations, the normal real time Feynman rules can still be used to give correct results. In this talk, I shall be concerned with the Real Time Formalism (RTF) of equilibrium thermal field theory as described using path integral techniques[1, 2, 3]. In particular, I would like to examine whether or not the partition function factorises into two pieces when using the RTF. This question is crucial to the RTF as it is precisely this factorisation which allows us to describe thermal effects in this formalism using thermal field doublets. Without factorisation we are forced to consider all the real time contour in closer detail[4] or to use another real time contour[4, 5]. I would like to begin by giving a brief description of what we mean by fac-torisation and the key reasons for our desire to split up the partition function in this way. To do this I shall use a single scalar field as a simple example. The generating functional of thermal Green functions is given by Z[J] = exp − ı C V [−ı δ δJ ] .Z 0 [J] Z 0 [J] = exp −ı 2 C dt C dt ′ J(t)∆ C (t − t ′)J(t ′) (1) where ∆ C (t − t ′) satisfies (2 + m 2)∆ C (t, t ′) = −δ C (t − t ′) subject to the KMS condition [6], and V [φ] is the interaction potential. I have suppressed spatial indices for notational convenience. Thermodynamic information may be obtained from Z[J = 0] which is the partition function. The curve, C in the complex time plane is the path associated with the Real Time Formalism (see figure 1).