It is an old idea to replace averages of observables with respect to a complex weight by expectation values with respect to a genuine probability measure on complexified space. This is precisely what ...one would like to get from complex Langevin simulations. Unfortunately, these fail in many cases of physical interest. We will describe method of deriving positive representations by matching of moments and show simple examples of successful constructions. It will be seen that the problem is greatly underdetermined.
The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density ...are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.
It is an old idea to replace averages of observables with respect to a complex weight by expectation values with respect to a genuine probability measure on complexified space. This is precisely what ...one would like to get from complex Langevin simulations. Unfortunately, these fail in many cases of physical interest. We will describe method of deriving positive representations by matching of moments and show simple examples of successful constructions. It will be seen that the problem is greatly underdetermined.
A local transformation from fermionic operators to spin matrices is proposed and studied in this work. For this purpose, a system of fermions on a lattice is considered and one applies the scheme to ...replace the fermionic variables with spin matrices, while the transformation relates only those fermionic/spin operators which are assigned to nearby lattice sites. In one dimension, this proposal yields the same result as the well-known Jordan-Wigner transformation, while not being restricted to \(d=1\) dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes \(3\times3\), \(4\times3\) and \(4\times4\). Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.
In the paper, a simple model of alpha decay with Dirac delta potential is studied. The model leads to breakdown of the exponential decay and to power law behavior at asymptotic times. Time dependence ...of the survival probability of the particle in the potential well is analyzed numerically with two methods: integration of Green's function representation and numerical solution of the time-dependent Schr\"odinger equation. In particular, finite depth potential wells and behavior between the exponential and power law regimes, which are situations that could not be described in detail analytically, are studied. The numerical results confirm power law with exponent n = 3 after the turnover into the non-exponential decay regime. Moreover, the constructive and destructive interference is observed in the intermediate stage of the process. The simple alpha decay model is compared to the results of Rothe- Hintschich-Monkman experiment which was the first experimental proof of violation of the exponential law.
It is an old idea to replace averages of observables with respect to a
complex weight by expectation values with respect to a genuine probability
measure on complexified space. This is precisely what ...one would like to get
from complex Langevin simulations. Unfortunately, these fail in many cases of
physical interest. We will describe method of deriving positive representations
by matching of moments and show simple examples of successful constructions. It
will be seen that the problem is greatly underdetermined.
Contrary to the common wisdom, local bosonizations of fermionic systems exist in higher dimensions. Interestingly, resulting bosonic variables must satisfy local constraints of a gauge type. They ...effectively replace long distance exchange interactions. In this work we study in detail the properties of such a system which was proposed a long time ago. In particular, dependence of the constraints on lattice geometry and fermion multiplicity is further elaborated and is now classified for all two dimensional, rectangular lattices with arbitrary sizes. For few small systems the constraints are solved analytically and the complete spectra of reduced spin hamiltonias are shown to agree with the original fermionic ones. The equivalence is extended to fermions in an external Wegner \(Z_2\) field. It is also illustrated by an explicit calculation for a particular configuration of Wegner variables. Finally, a possible connection with the recently proposed web of dualities is discussed.
The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density ...are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.
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