Summary
The light-front Hamiltonian formulation for the scalar field theory described in the continuum, which contains also a constraint equation, is constructed and compared with the equal-time formulation. In the two-dimensional case the mass renormalization condition and the renormalized constraint equation are shown to contain the information necessary to describe the phase transition in the ϕ4 theory and it is found to be of the second order. We argue that the same result is obtained also in the conventional equal-time formulation.
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From the momentum space expansion of ϕ the free (light front) propagator is derived to beiG 0(x;0)≡<0|T(φ(x,τ)φ(0,0))|0>=∫(dk4πk)θ(τ)exp[−i(kx+ɛ k τ)]+θ(−τ)· exp[i(kx+ɛ k τ)]], where 2kɛ k =M 20 . On using the well-known integral representation of ϑ(τ) and the identity [ϑ(k)+ϑ(−k)]=1, true in the sense of distribution theory, it may be rewritten as ∫∫(dk +dk −(2π)2)[i(2k + k −−M 20 +iɛ) exp[−i(k + x −+k − x +)]. Herek ± are now dummy variables taking values from −∞ to ∞ and theintegration over k − is understood to be performed first. This is exactly like the discussion in the equal-time case where we find that the integral over the time component is understood to be done first (e.g., see alsoSchweber S. S.,Relativistic Quantum Field Theory (Row, Peterson and Co., New York, N.Y.) 1961, p. 443). In order to arrive at (3.2) we make a change of variables from (k −,k +) to (k −,k +) to (k 0=(k −+k +)/√2,k +) so that the integration over the new variablek 0 is now to be performed first. From (2k + k −−M 20 +iɛ)−1=[2k (√2k 0−k +)−M 20 +iɛ)]−1, it follows that the pole in the variablek 0 is situated at [k++M 20 (2k +)−iɛ/(2k +)]/√2. It lies in the fourth quadrant below the real axis in the complexk 0 plane fork +>0 and in the second quadrant above the real axis fork +<0. Hence the (usual) Wick rotation is allowed to be performed in the variablek 0. We may then make another change of variables (k 0),k + to (k 0,k 1=[√2k +−k 0]) so that 2k + k −= (k 0 2−k 1 2) Upon performing a Wick rotation ink 0 we obtain the Euclidean space integral used in the text.
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Srivastava, P.P. Light-front quantization and the phase transition in (ϕ4)2 theory. Nuov Cim A 108, 35–45 (1995). https://doi.org/10.1007/BF02814855
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DOI: https://doi.org/10.1007/BF02814855