Calculation of multi-time correlation functions in dissipative quantum φ^4 theory

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Αλατάς, Παναγιώτης
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In quantum field theories, Feynman diagrams depict visually the interaction of elementary particles and involve closed loops, corresponding to integration over all possible combinations of energy and momentum of the virtual particles traveling in the space-time continuum, leading often to infinities. To treat these infinities, renormalization of the parameters of the theory is required. Renormalization specifies relationships between parameters in the theory whose values at large length scales are different from those at short length scales. Behind renormalization lies the problem of the proper elimination of degrees of freedom in order to bridge the gap between different length and time scales. Nonequilibrium thermodynamics offers an alternative approach to the problem of the time evolution of an open quantum system through a time-evolution master equation for the density matrix of the system, which includes reversible and irreversible contributions. In particular, irreversibility introduces dissipative smoothing mechanisms for the terms responsible for the infinities, providing the proper regularization of the theory. The objective of the present doctoral dissertation is the implementation of the dissipative quantum field theory (DQFT) developed by Prof. Dr. ‪Hans Christian Öttinger to calculate the perturbative expansions up to third order in the interaction parameter of the two- (propagator) and four-point (effective interaction vertex) correlation functions of the scalar theory with quartic interactions (φ^4 theory) in the limit of zero temperature, as well as to extract the respective Feynman diagrams where (in the context of DQFT) collision events have a well-defined arrow of time. To achieve this, we developed a powerful symbolic code which allowed us to organize and simplify the outcome of the commands performing the calculation of terms contained in each function within a given order so that to lead to a compact final expression, thus rendering the identification of the relevant Feynman diagrams more transparent. The use of symbolic computation was motivated by the fact that the perturbative DQFT in d dimensions (d denotes the number of space dimensions) is considerably more complicated than the usual calculations in D = d + 1 dimensions (D the number of space + time dimensions) of the Lagrangian quantum field theory, generating a huge number of terms with increasing order of perturbation. As expected, the resulting expressions for the propagator and the effective interactive vertex contain terms corresponding to zeroth-, first-, second- and third-order contributions. The terms up to second order have already been reported by Prof. Dr. Hans Christian Öttinger. In this Thesis, we computed the third-order contributions to both of them in terms of connected and unconnected Feynman diagrams and studied their topology. In the limit of vanishing dissipation (zero friction coefficient), the final expressions for the two correlation functions analyzed in this Thesis coincide with those obtained from the Lagrangian quantum field theory. This is achieved by integrating out the time or energy component in the D-dimensional integrals of the Lagrangian approach to obtain the d-dimensional integrals of the dissipative approach. The final expression for the propagator up to third order is regularized with two different schemes, one of them being the so-called “on shell” scheme, thus obtaining fully convergent results in d = 3 dimensions. Even more important, by combining our results for the second- and third-order contributions at the long-time or low-energy limit, we confirm that the value of the critical coupling constant λ* as a function of space dimensionality d is correctly reproduced, which demonstrates the power of the new, dissipative approach to quantum field theory and the correct implementation (in its entirety) of the powerful symbolic code developed in this Thesis. The corresponding dissipative result for the effective interaction vertex is expressed in terms of a parameter describing the level of energy conservation violation. A compact and rather friendly expression is obtained by assuming that the external legs are amputated and the mass is equal to zero, demonstrating the finiteness of the terms appearing in the vertex, in the limit of vanishing friction, which is a great outcome of the dissipative approach followed. Moreover, many striking similarities are pointed out and discussed with the corresponding Lagrangian result at the same order of perturbation.
Nonequilibrium thermodynamics, Feynman diagrams, Symbolic computation, Propagator, Quantum field theories