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Systematic method for unification of various field theories in a two-dimensional classical $\phi^4$ field theory

Mohammad Hossein Zarei, Yahya Khalili

Published Aug 3, 2016• 0 citations
OpenAlex

Abstract

Although creating a unified theory in Elementary Particles Physics is still an open problem, there are a lot of attempts for unifying other fields of physics. Following such unifications, we regard a two dimensional (2D) classical $\Phi^{4}$ field theory model to study several field theories with different symmetries in various dimensions. While the completeness of this model has been already proved by a mapping between statistical mechanics and quantum information theory, here, we take into account a fundamental systematic approach with purely mathematical basis to re-derive such completeness in a general manner. Due to simplicity and generality, we believe that our method leads to a general approach which can be understood by other physical communities as well as quantum information theorists. Furthermore, our proof of the completeness is not only a proof-of-principle, but also an interesting algorithmic proof. We consider a discrete version of a general field theory as an arbitrary polynomial function of fields which is defined on a lattice with arbitrary dimension. Then we give a few specific transformations on such field theories to convert them to a classical $\phi^4$ field theory on a 2D square lattice with the same Partition function. We also extend the results to the $U(1)$ lattice gauge theories in arbitrary dimensions.

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Systematic method for unification of various field theories in a two-dimensional classical ϕ4\phi^4ϕ4 field theory
Although creating a unified theory in Elementary Particles Physics is still an open problem, there are a lot of attempts for unifying other fields of physics. Following such unifications, we regard a two dimensional (2D) classical Φ4\Phi^{4}Φ4 field theory model to study several field theories with different symmetries in various dimensions. While the completeness of this model has been already proved by a mapping between statistical mechanics and quantum information theory, here, we take into account a fundamental systematic approach with purely mathematical basis to re-derive such completeness in a general manner. Due to simplicity and generality, we believe that our method leads to a general approach which can be understood by other physical communities as well as quantum information theorists. Furthermore, our proof of the completeness is not only a proof-of-principle, but also an interesting algorithmic proof. We consider a discrete version of a general field theory as an arbitrary polynomial function of fields which is defined on a lattice with arbitrary dimension. Then we give a few specific transformations on such field theories to convert them to a classical ϕ4\phi^4ϕ4 field theory on a 2D square lattice with the same Partition function. We also extend the results to the U(1)U(1)U(1) lattice gauge theories in arbitrary dimensions.
Systematic unification of the partition functions of various field theories in a two dimensional classical ϕ4\phi^4ϕ4 model
We show that the partition functions of several field theories with different symmetries and in various dimensions are equal to the partition function of a two-dimensional (2D) classical ϕ4\phi^4ϕ4 field theory model. Although the completeness of 2D ϕ4\phi^4ϕ4 model has been already proved by introducing a quantum formalism for the partition functions, in this paper we study it in a fundamental systematic approach with purely mathematical basis to re-derive such completeness in a general manner. Due to generality, such re-derivation will lead to more clarification of the result for other communities of physics as well as quantum information theorists. Furthermore, our proof of the completeness has an important advantage that is a step-by-step proof which is supported by some graphical transformations. Specifically, we consider discrete version of field theories on lattices with arbitrary dimensions where they are defined as arbitrary polynomial functions of fields with different symmetries. Then we give a set of simple graphical transformations on such field theories to convert them to a classical ϕ4\phi^4ϕ4 field theory on a 2D square lattice with the same partition function.
In our recent work [Phys. Rev. Lett. 102, 230502 (2009)] we showed that the partition function of all classical spin models, including all discrete standard statistical models and all Abelian discrete lattice gauge theories (LGTs), can be expressed as a special instance of the partition function of a 4-dimensional pure LGT with gauge group Z_2 (4D Z_2 LGT). This provides a unification of models with apparently very different features into a single complete model. The result uses an equality between the Hamilton function of any classical spin model and the Hamilton function of a model with all possible k-body Ising-type interactions, for all k, which we also prove. Here, we elaborate on the proof of the result, and we illustrate it by computing quantities of a specific model as a function of the partition function of the 4D Z_2 LGT. The result also allows one to establish a new method to compute the mean-field theory of Z_2 LGTs with d > 3, and to show that computing the partition function of the 4D Z_2 LGT is computationally hard (#P hard). The proof uses techniques from quantum information.
The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical [Formula: see text] filed theory whose completeness has been proved in [V. Karimipour and M. H. Zarei, Phys. Rev. A 85 (2012) 032316]. We give a new and general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical [Formula: see text] model. To this end, we start from various classical field theories containing models on arbitrary lattices and also [Formula: see text] lattice gauge theories. Then we convert them to a new classical field model on a nonplanar bipartite graph with imaginary kinetic terms. After that, we show that any polynomial function of the field in the corresponding Hamiltonian can approximately be converted to a [Formula: see text] term by adding enough numbers of vertices to the bipartite graph. In the next step, we give a few graphical transformations to convert the final nonplanar graph to a 2D rectangular lattice. We also show that the number of vertices which should be added grows polynomially with the number of vertices in the original model.
We have recently shown that the partition function of any classical spin model, including all discrete standard statistical models and all Abelian discrete lattice gauge theories, can be expressed as the specific instance of the partition function of a four dimensional Ising lattice gauge theory. This unifies models with apparently very different features (global vs. local symmetries, many body interactions, different dimensions, etc) in a single complete model. The proof of the result uses techniques from quantum information. In this talk we review these mappings and discuss recent developments, including quantum algorithms to approximate the partition function.
The partition function of all classical spin models, including all discrete standard statistical models and all Abelian discrete lattice gauge theories (LGTs), is expressed as a special instance of the partition function of the 4D Z2 LGT. This unifies all classical spin models with apparently very different features in a single complete model. This result is applied to establish a new method to compute the mean-field theory of Abelian discrete LGTs with d > or = 4, and to show that computing the partition function of the 4D Z2 LGT is computationally hard (#P hard). The 4D Z2 LGT is also proved to be approximately complete for Abelian continuous models. The proof uses techniques from quantum information.
We formulate a quantum formalism for the statistical mechanical models of discretized field theories on lattices and then show that the discrete version of ${\ensuremath{\varphi}}^{4}$ theory on 2D square lattice is complete in the sense that the partition function of any other discretized scalar field theory on an arbitrary lattice with arbitrary interactions can be realized as a special case of the partition function of this model. To achieve this, we extend the recently proposed quantum formalism for the Ising model [M. Van den Nest, W. Dur, and H. J. Briegel, Phys. Rev. Lett. 98, 117207 (2007)] and its completeness property [M. Van den Nest, W. Dur, and H. J. Briegel, Phys. Rev. Lett. 100, 110501 (2008)] to the continuous variable case.