Renormalization Group And Fixed Points In Quantum Field Theory Pdf
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Free theory and Wick's theorem.
In theoretical physics , the term renormalization group RG refers to a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. A change in scale is called a scale transformation. The renormalization group is intimately related to scale invariance and conformal invariance , symmetries in which a system appears the same at all scales so-called self-similarity.
Due to the wide extent of this research field, a complete listing of main contributions is impractical. We mention here few examples that are representative for the topic of this work: anomalous diffusion and Levy statistics in Hamiltonian phase space [12,13], non-differentiability of Feynman paths [14,15], application of fractional Brownian motion in quantum field theory , vacuum fluctuations, chaotic maps and stochastic quantization in particle physics , mass generation in the lepton sector due to period doubling transition to chaos , quantum Brownian motion , fractional dynamics and origin of the fine-structure constant , Cantorian space-time and the topological foundation of coupling and mass spectra in the Standard Model .
Drawing on recent results regarding renormalization group in the presence of quantum fluctuations , our work suggests that the random topology of space-time is not an exclusive attribute of the Planck scale but an inherent manifestation of stochastic dynamics near any fixed point of the underlying field theory.
The paper is organized as follows: Section 2 introduces the concept of random observation scale from arguments related to equilibrium statistical mechanics. Taking the u 4 theory as a benchmark model, Section 3 examines the behavior of the RG solution near the unique fixed point g!
The stochastic character of the field exponent is analyzed in Section 4. Section 5 investigates the temporal evolution of the u field from a statistical mechanics perspective. Connection of space-time coordinates to fractal objects having random dimension is discussed in Section 6. Results are summarized in Section 7. The random observation scale: a statistical mechanics argumentClassical statistical mechanics of systems at thermal equilibrium asserts that energy or energy and number of particles in thermodynamic ensembles are subject to incessant fluctuations.
Consider, for example, a system enclosed in a heatbath at constant finite temperature T a ''canonical ensemble''. Fluctuations vanish for macroscopic systems in the limit N! In contrast, fluctuations survive in canonical ensembles comprising low-dimensional classical or quantum systems such as dilute Bose gases. In the same context, we recall that quantum fields are objects with manifest statistical properties owing to the cascade of virtual processes that occur during propagation and interaction.
We also recall that a basic requirement of any realistic quantum field theory is renormalizability . To render all computations finite, a well-established regularization procedure needs to be implemented.
This sets the ''coarse-graining'' scale and the resolution at which the underlying physics is probed. From these arguments it follows that, if the average energy of the quantum field system hEi sets its temperature, then l is expected to undergo continuous fluctuations about hEi.
There are no preferential values in this random occurrence. Let l be defined inside a range bounded by l M and l m. During s we assume that the system randomly samples all available energy scales contained in the range. A comment on 2 is now in order. Dimensional consistency requires the right hand side of 2 to be expressed in scalar form. According to this interpretation, the time window s 0 and energy scale l 0 are statistically conjugate variables.
The natural outcome of this conjecture is that time behaves as a stochastic variable. As known, coefficient functions are specific for each field theory. In the neighborhood of these points i. It can be seen that, in general, both field and mass solutions display a N -fold multiplicity and acquire properties of continuous random variables.
Let m 0 i ; u 0 i and m 0 f ; u 0 f designate two random sets of initial and final mass and field states. Stochastic nature of field scalingA well-known property of critical phenomena is that, near transition points, all relevant variables scale in a similar fashion with the control parameter.
The scaling behavior is defined by a set of fixed exponents that are deterministic in nature and dependent on the dimensionality of the system [5,7,8].
In contrast, we are going to show in this section that the field exponent c 0 acquires a stochastic character due to the postulated randomness of the observation scale l 0. Upon independent scaling of the non-dimensional space-time coordinates according to 6 , it can be shown that the field propagator changes as  Let the initial state be fixed and the final state randomly vary. Up to an additive constant 22 may be written as4 E. Related Papers. Derivation of the fine structure constant using a fractional dynamics approach.
By Ervin Goldfain. Fractional dynamics, Cantorian space—time and the gauge hierarchy problem. Local scale invariance, Cantorian space—time and unified field theory. Hypersingular integral equations, waveguiding effects in Cantorian Universe and genesis of large scale structures. On the intuitionistic fuzzy topological spaces. By Reza Saadati.
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WS18 Quantum Field Theory
It seems that you're in Germany. We have a dedicated site for Germany. This Brief presents an introduction to the theory of the renormalization group in the context of quantum field theories of relevance to particle physics. Emphasis is placed on gaining a physical understanding of the running of the couplings. The Wilsonian version of the renormalization group is related to conventional perturbative calculations with dimensional regularization and minimal subtraction. An introduction is given to some of the remarkable renormalization group properties of supersymmetric theories.
This Brief presents an introduction to the theory of the renormalization group in the context of quantum field theories of relevance to particle physics. Emphasis is placed on gaining a physical understanding of the running of the couplings. The Wilsonian version of the renormalization group is related to conventional perturbative calculations with dimensional regularization and minimal subtraction. An introduction is given to some of the remarkable renormalization group properties of supersymmetric theories. Modern Optical Spectroscopy gives clear explanations of the theory of optical spectroscopic phenomena and shows how these ideas are used in modern molecular and cellular biophysics and biochemistry.
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Claren- don Press the introduction. Wilson-Fisher fixed point on the site catamountconnections.org
Renormalization Group and Fixed Points
Don't have an account? Up to now, we have mainly discussed the IR behaviour of field theories. This chapter uses RG equations to characterize instead the large momentum behaviour of renormalized field theories. This assumes implicitly that a universal large momentum physics, that is, a property of the continuum, can be defned. This implies also the existence of a crossover scale between low and large momentum physics.
Anselmi Theories of gravitation. Differential geometry. General relativity.
This Brief presents an introduction to the theory of the renormalization group in the context of quantum field theories of relevance to particle physics. Emphasis is placed on gaining a physical understanding of the running of the couplings. The Wilsonian version of the renormalization group is related to conventional perturbative calculations with dimensional regularization and minimal subtraction. An introduction is given to some of the remarkable renormalization group properties of supersymmetric theories.
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The lecture is aimed at master students with an interest in theoretical physics. It is a crucial preparation for a master thesis in theoretical particle physics. The quantum field theory concepts discussed are however more widely applicable. The focus here will be on methods, rather than on phenomenology as compared to the 'Theoretical particle physics' course. Strongly advised for students who have not attended the "Relativity, Particles, Fields'' course at TUM in the summer or wish to refresh their theoretical physics background.
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