Path integral on star graph

Abstract

In this paper we study path integral for a single spinless particle on a star graph with N edges, whose vertex is known to be described by U (N ) family of boundary conditions. After carefully studying the free particle case, both at the critical and off-critical levels, we propose a new path integral formulation that correctly captures all the scale-invariant subfamily of boundary conditions realized at fixed points of boundary renormalization group flow. Our proposal is based on the folding trick, which maps a scalar-valued wave function on star graph to an N -component vector-valued wave function on halfline. All the parameters of scale-invariant subfamily of boundary conditions are encoded into the momentum independent weight factors, which appear to be associated with the two distinct path classes on half-line that form the cyclic group Z 2 . We show that, when bulk interactions are edge-independent, these weight factors are generally given by an N -dimensional unitary representation of Z 2 . Generalization to momentum dependent weight factors and applications to worldline formalism are briefly discussed.

Path integral junctions

“…After proving this statement, we will rederive the scale-invariant boundary conditions by using unitary representations. The proof is almost parallel to that presented in the previous work [7] such that we here discuss only the composition rule. By substituting the ansatz (3.2), the left hand side of (2.4b) becomes n,m∈Z…”

“…Noting that every element of Z 2 * Z 2 is given by an alternating product of P 0 and P ℓ , we immediately see that the map Z 2 * Z 2 → D ∞ given by (P 0 P ℓ ) n → T n and (P 0 P ℓ ) n P 0 → T n R is an isomorphism. The second point to note is that N -dimensional unitary representation of Z 2 is just given by an N × N hermitian unitary matrix [7]. Therefore, assigning two distinct hermitian unitary matrices to P 0 and P ℓ , we obtain the following N -dimensional unitary representations of D ∞ :…”

A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalism

“…A Proof of the path-integral formula (33) Following the ideas presented in [30,31], in this section we show that the formula (33) satisfies the properties ( 27)-( 30) if is a one-dimensional unitary representation of and if R fulfills the conditions (34)-(38). Below we prove these four properties separately.…”

Thermal fluctuations of an interface near a contact line

“…Since ψ −φ (h) can be obtained from ψ φ (h) by means of a unitary transformation [106], the linearly independent eigenstates are those for φ > 0. Furthermore, if (and only if) θ > 0, there exists an eigenstate having a negative energy eigenvalue: the bound eigenstate corresponding to ǫ = ǫ 0 ≡ −θ 2 /2 is given by [106] ψ 0 (h) = 1+sgn θ 2 2θ ℓ e − θh ℓ , sgn θ being the sign of θ. Note that the bound state is denoted by ψ 0 ≡ ψ ǫ=ǫ0 and should not be confused with ψ φ=0 ≡ ψ ǫ=0 , the latter being identically vanishing.…”