2
Relativistic Point Particle
As briefly explained in the introduction, the BV formalism plays the central role in
the construction of a gauge invariant action for string field theory. By now, there
are many excellent texts on the BV formalism. We will thus not present a detailed
account. However, the construction of a BV action in string field theory is rather
different from that in gauge theories. In some ways it goes backwards! Indeed, in the
original BV construction, the starting point is a gauge invariant action (or boundary
condition). The gauge theory is then extended by addition of further fields (ghost,
ghosts for ghosts, etc.) and anti-fields. The action gets extra terms which contain new
fields and anti-fields. Expanding the so obtained extended action around a critical
point naturally induces an L ∞ -algebra structure. In contrast, in string field theory,
the initial action or boundary condition is generically unknown. Nevertheless,
starting from the world-sheet description which is a convenient formulation of string
theory, and decomposing the scattering amplitudes into irreducible components, one
directly constructs the perturbative expansion of the BV action around a critical
point with all anti-fields included.
It turns out that a simplified version of the algebraic structure described in the
introduction is already present for point particles. We will therefore begin with
the construction of the BV action for a spinless particle starting from the worldline formalism. Of course, for that system a consistent field theory can easily be
constructed by other means as it is done in the standard textbooks of quantum field
theory. However, for illustration of the formalism at work in string theory, including
the algebraic structure and its operadic description, this approach seems appropriate.
2.1
BV Action for the Point Particle
We consider a single point particle propagating on a non-compact manifold M with
a pseudo-Riemannian metric g. The world-line of the particle defines a parametrized
curve φ : [τ i , τ f ] → M. The basic object in our construction of a space-time action
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_2
9
Relativistic Point Particle
As briefly explained in the introduction, the BV formalism plays the central role in
the construction of a gauge invariant action for string field theory. By now, there
are many excellent texts on the BV formalism. We will thus not present a detailed
account. However, the construction of a BV action in string field theory is rather
different from that in gauge theories. In some ways it goes backwards! Indeed, in the
original BV construction, the starting point is a gauge invariant action (or boundary
condition). The gauge theory is then extended by addition of further fields (ghost,
ghosts for ghosts, etc.) and anti-fields. The action gets extra terms which contain new
fields and anti-fields. Expanding the so obtained extended action around a critical
point naturally induces an L ∞ -algebra structure. In contrast, in string field theory,
the initial action or boundary condition is generically unknown. Nevertheless,
starting from the world-sheet description which is a convenient formulation of string
theory, and decomposing the scattering amplitudes into irreducible components, one
directly constructs the perturbative expansion of the BV action around a critical
point with all anti-fields included.
It turns out that a simplified version of the algebraic structure described in the
introduction is already present for point particles. We will therefore begin with
the construction of the BV action for a spinless particle starting from the worldline formalism. Of course, for that system a consistent field theory can easily be
constructed by other means as it is done in the standard textbooks of quantum field
theory. However, for illustration of the formalism at work in string theory, including
the algebraic structure and its operadic description, this approach seems appropriate.
2.1
BV Action for the Point Particle
We consider a single point particle propagating on a non-compact manifold M with
a pseudo-Riemannian metric g. The world-line of the particle defines a parametrized
curve φ : [τ i , τ f ] → M. The basic object in our construction of a space-time action
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_2
9