Wave functionals of free real and complex scalar fields on a dimensional lattice
We obtain wave functionals of free real and complex scalar fields on a dimensional lattice by explicitly calculating the path integral for transition from one field configuration to another. The obtained expressions are useful for cross-checking quality of approximations schemes used to study self-interacting fields on the lattice.
In cases when the perturbative methods are not applicable a powerful method to study properties of a system under consideration is to put it on the lattice and study its properties numerically. Examples include a real scalar field with double well potential,
and a system of two scalar fields with Mexican hat potential,
in the limit of very small but nonzero . To cross check the quality of the numerical approximations one can use the same numerical method to analyze models with known analytical solutions. For (1) the cross check can be done using the lattice version of a free massive scalar field. On a dimensional Euclidean lattice of a spatial size its Lagrangian reads
where , , is the number of sites,
and . As can be inferred from (3) we deal with a system of coupled oscillators . The finiteness of the space introduces a periodicity condition and therefore . The Lagrange functional is given by . To diagonalize it we introduce Fourier-transforms of according to
where . The requirement that be real-valued implies that so that the number of independent degrees of freedom is again equal to K. The Jacobian of this transformation is unity. In the momentum representation the Lagrange function is given by a sum of Lagrange functions of independent oscillators,
where is the lattice version of the relativistic energy-momentum relation.
One of the ways to extract energy spectrum and wave functionals of (6) is to use spectral representation of the transition amplitude from one field configuration, , to another one, . The transition amplitude can be also computed using the path integral. Representing the paths as a sum of the classical trajectory and fluctuations we can write the latter in the form
Solving the (Euclidean version of the) Lagrange equations of motion we find
where . The classical action then reads
The Lagrange function of the fluctuations also has the form (6) but with the boundary conditions . The boundary conditions take into account that (by definition) the fluctuations vanish at and . To calculate action of the fluctuations it is convenient to split the time interval from to into small intervals and Fourier-transform with respect to time. The boundary conditions for the fluctuations imply that the cosine term of the Fourier-transformation vanishes and we are left with
where . Jacobian of this transformation is again unity. Using (10) we can write the action of the fluctuations in a compact form
Finally, the measure in the path integral (7) is given by
The symbol denotes independent degrees of freedom. As has been mentioned above, due to the periodicity condition components of and are related by . This implies in particular that is real-valued. Taking in addition into account the periodicity of the eigenvalues (12) it is convenient to choose the independent degrees of freedom in such a way that, for instance, and , where . This definition implies also that . Rewritten in terms of , the action on the classical trajectory (Wave functionals of free real and complex scalar fields on a dimensional lattice) takes the form
For the action of the fluctuations we find
Since all are positive the surfaces of constant action are ellipses in the dimensional space. The result of the Gaussian integration is proportional to the product of . The product over is given by
where as . Combining (16) and (Wave functionals of free real and complex scalar fields on a dimensional lattice) and taking the limit we finally find for the transition amplitude
The first factor on the right-hand side of (Wave functionals of free real and complex scalar fields on a dimensional lattice) is the fluctuation determinant and the remaining two factors corresponds to the classical trajectory.
Making use of a summation formula for Hermite polynomials Kleinert (2004) we can obtain the spectral representation of (Wave functionals of free real and complex scalar fields on a dimensional lattice)
The wave functionals are given by
and the energy levels by
The vacuum corresponds to for all . In this case the second line in (Wave functionals of free real and complex scalar fields on a dimensional lattice) is equal to unity. Performing inverse Fourier transformation we find for the wave functional of the vacuum state
where is the original scalar field on the lattice. The normalization factor and the matrix are given by
The fact that depends on the difference of and reflects translational invariance of the vacuum. Using Fourier representation of the delta-function on the lattice and the definition of a function of an operator we can represent it in the form . It is identical to the form that can be obtained for the real scalar field using the Schrödinger representation Jackiw (1987).
For (2) the cross check can be done using the lattice version of a system of two free massive scalar fields with equal masses, which is equivalent to a free complex scalar field. On a dimensional Euclidean lattice its Lagrangian reads
Because the two field degrees of freedom are independent the resulting transition amplitude is given by a product of the two amplitudes (Wave functionals of free real and complex scalar fields on a dimensional lattice). Introducing the radial and angular components, and , and using the relation Kleinert (2004) we can represent it in the form
Making use of the Hille-Hardy formula Grosche and Steiner (1995) we can extract from (Wave functionals of free real and complex scalar fields on a dimensional lattice) the energy spectrum of the system,
as well as the form of the corresponding wave functionals,
where are the generalized Laguerre polynomials. The vacuum corresponds to for all . In this case the second and third lines in (Wave functionals of free real and complex scalar fields on a dimensional lattice) are equal to unity and the wave functional is cylindrically symmetric in the Fourier space. Performing inverse Fourier transformation we find for the wave functional of the vacuum
where the radial and angular variables are related to the field components by and . The fact that the wave function depends on the difference of the field phases indicates that the vacuum is cylindrically symmetric.
To summarize, we have calculated energy spectra and wave functionals of free real and free complex scalar fields on a 1+1 dimensional space using the path integral method. The obtained expressions are useful for cross-checking quality of approximations schemes that are used to study self-interacting fields on the lattice.
- Kleinert (2004) H. Kleinert, Path integrals in quantum mechanics, statistics, polymer physics, and financial markets (World Scientific, 2004), 3rd edition, chapter 2.9.
- Jackiw (1987) R. Jackiw (1987), Schroedinger picture analysis of boson and fermion quantum field theories.
- Grosche and Steiner (1995) C. Grosche and F. Steiner, J.Math.Phys. 36, 2354 (1995).