2.1 BV Action for the Point Particle
13
The action of Q on V is unambiguously determined by the representation (2.8)
and (2.9). Furthermore, Qψ = 0 enforces the physical or on-shell condition.
More precisely, the physical states are usually identified with H | deg 0 , where
H = coh(Q, V ), but in this book we will sometimes ignore the restriction to deg
0 and refer to physical states simply as to elements of H and use sometimes the
notation V phys or V P for it.
Given our construction of V above, ψ ∈ V is physical or on shell if H ψ 0 = 0.
Note that, since H = [Q, b], the world-line Hamiltonian H for this dynamical
system is Q-exact.
Returning to the path integral, for a given interval Δτ = τ f − τ i , the kernel
K(Δτ, φ 1 , φ 0 ) has the physical interpretation of the quantum-mechanical propagation in proper time between initial and final state of the particle by integrating the
kernel against the wave functions ψ(φ, c) as
K(Δτ, ψ 2 , ψ 1 ) =
dc dφ 2 dφ 1 ¯
ψ 2 (φ 2 , c)ψ 1 (φ 1 , c) K(Δτ, φ 2 , φ 1 ) .
(2.11)
Here, the integral
dc over the constant c-ghost arises from fixing the invariance of
the world-line path integral measure under proper-time translations. For Δτ → 0,
the functional integral kernel (2.6) reduces to a delta function 1
δ(φ 2 − φ 1 )
(2.12)
thus (2.11) defines a degree −1 symplectic form
2 , ψ 1 ≡ K(0, ψ 2 , ψ 1 ) =
dcdφ ¯
ψ 2 (φ, c)ψ 1 (φ, c) .
(2.13)
Note that, due to the presence of the c-zero mode, the paring is between subspaces of
V with different degree. This may seem counter intuitive but it is, in fact, a common
feature in BV quantization.
We now have all ingredients needed to define an action functional on V ,
S[ψ] =
1
2
ψ, Q ψ .
(2.14)
This action reproduces the correct on-shell, or physical state condition from the
variational principle for ψ when evaluated at degree 0.
Interactions between scalar particles can be included by considering the propagation of the first particle in the potential of the second as in Fig. 2.1. If we denote
by ψ 2 (φ) the wave function of the second particle, then the world-line of the first
particle couples to the latter through ψ 2 (φ(s)) where s is the proper time of the first
1 Strictly speaking this holds only for Riemannian signature on M and should be suitably defined
with an ii prescription as known from quantum field theory textbooks.
13
The action of Q on V is unambiguously determined by the representation (2.8)
and (2.9). Furthermore, Qψ = 0 enforces the physical or on-shell condition.
More precisely, the physical states are usually identified with H | deg 0 , where
H = coh(Q, V ), but in this book we will sometimes ignore the restriction to deg
0 and refer to physical states simply as to elements of H and use sometimes the
notation V phys or V P for it.
Given our construction of V above, ψ ∈ V is physical or on shell if H ψ 0 = 0.
Note that, since H = [Q, b], the world-line Hamiltonian H for this dynamical
system is Q-exact.
Returning to the path integral, for a given interval Δτ = τ f − τ i , the kernel
K(Δτ, φ 1 , φ 0 ) has the physical interpretation of the quantum-mechanical propagation in proper time between initial and final state of the particle by integrating the
kernel against the wave functions ψ(φ, c) as
K(Δτ, ψ 2 , ψ 1 ) =
dc dφ 2 dφ 1 ¯
ψ 2 (φ 2 , c)ψ 1 (φ 1 , c) K(Δτ, φ 2 , φ 1 ) .
(2.11)
Here, the integral
dc over the constant c-ghost arises from fixing the invariance of
the world-line path integral measure under proper-time translations. For Δτ → 0,
the functional integral kernel (2.6) reduces to a delta function 1
δ(φ 2 − φ 1 )
(2.12)
thus (2.11) defines a degree −1 symplectic form
2 , ψ 1 ≡ K(0, ψ 2 , ψ 1 ) =
dcdφ ¯
ψ 2 (φ, c)ψ 1 (φ, c) .
(2.13)
Note that, due to the presence of the c-zero mode, the paring is between subspaces of
V with different degree. This may seem counter intuitive but it is, in fact, a common
feature in BV quantization.
We now have all ingredients needed to define an action functional on V ,
S[ψ] =
1
2
ψ, Q ψ .
(2.14)
This action reproduces the correct on-shell, or physical state condition from the
variational principle for ψ when evaluated at degree 0.
Interactions between scalar particles can be included by considering the propagation of the first particle in the potential of the second as in Fig. 2.1. If we denote
by ψ 2 (φ) the wave function of the second particle, then the world-line of the first
particle couples to the latter through ψ 2 (φ(s)) where s is the proper time of the first
1 Strictly speaking this holds only for Riemannian signature on M and should be suitably defined
with an ii prescription as known from quantum field theory textbooks.