Speaker
Description
We explore O(2N) scalar theory with fractional Laplacian, √(-∇²) in d-dimension and its Hamiltonain dynamics which is described by a Schrodinger type equation. This equation is a kind of current conservation equation, where one can define a current j(φ) of a probability P(φ), where φ is the scalar field. Naturally, Gibbs entropy S=-∫ [Dφ] P(φ)log P(φ) can be considered to explore the system. We realize that this Gibbs entropy of the O(2N) scalar theory with fractional Laplacian is matched with free energy of O(N) vector model in finite temperature, 1/β with chemical potnential, μ in d-dimension. The precise map between the stochastic fictitious time t and the inverse temperature β is β=2t. Therefore, the tmeperature dependence of the thermal O(N) vector model can be realized as a dynamics of time dependent solution satisfying Schrodinger type equation. This free energy is obtained by putting O(N) vector model in S¹× R_d, where S¹ is thermal circle with its periodicity β. To get d-dimensional theory, we sum up all possible frequencies on the circle(so called Matsubara frequency summation) which gives d-dimensional thermal partition function. We note that the nontrivial t-dependence appears beyond classical limit.
To take into account quantum effects, we solve the Hamiltonian dynamics by keeping ℏ corrections. The chemical potential is mediated by a parameter l such that μ=log l/β and so we call this l-deformation. This is related to the boundary condition of the Schrodinger equation. We also note that the two theoreis are not equivalent each other and we just check their correspondence in the level of one-loop determinant, i.e. zero point function in the note.
