24
2 Relativistic Point Particle
as a morphism from the cyclic L ∞ -algebra (H, ω| H , S) on the cohomology H to
the original L ∞ -algebra (V , ω, L). Formulas (2.36) and (2.37) suggest one possible
definition of a morphism in a category of cyclic L ∞ -algebras as a map (in general a
nonlinear one) between the underlying vector spaces compatible with the respective
cyclic L ∞ -structures. Obviously, a map respecting symplectic structures might
seem to be a too restrictive one, the dimension of the source vector space cannot be
bigger than that of the target vector space. A possible way out would be to consider
Lagrangian correspondences instead of maps. We will not need this generalization.
Hence, we will not comment on this any further.
The minimal model defined on the subspace of physical states is then quasiisomorphic to the L ∞ -algebra defined by the original action S on the space of
all states. Formula (2.35) for the tree-level scattering amplitudes/brackets of the
minimal model is the content of the homological perturbation lemma. It expresses
the change of the trivial differential on functions on the Q-cohomology H induced
by the perturbation of the BRST operator Q by the product ∗ on functions on V .
Finally, we would like to comment on the relation to the operadic formulation
of cyclic L ∞ -algebras in Part II of this book. The maps l n or equivalently, their
duals Q
(k)
φ 0
and indeed, the classical BV action S, define a cyclic L ∞ -algebra. This
is an algebra over the cobar construction of the cyclic commutative operad briefly
recalled at the beginning of Sect. 7.2 in Part II. The minimal model construction,
when expressed by formula (2.35), applies directly to any algebra over the cobar
construction of a general cyclic operad and even more generally to any algebra over
the Feynman transform of a general modular operad. In the form presented here
it also applies to representations of cobar constructions of properads and hence, in
particular, to IBL ∞ -algebras.
Further Reading
An intuitive pedagogical account of the quantization of the relativistic point particle
can be found in:
• A. M. Polyakov, “Gauge fields and strings”, Harwood, 1987.
A standard reference for BRST and BV quantization is:
• M. Henneaux and C. Teitelboim, “Quantization of gauge systems”, Princeton
University Press, Princeton 1992.
Other useful references include:
• S. Weinberg, “The quantum theory of fields Vol. II”, Cambridge University Press,
2005,
2 Relativistic Point Particle
as a morphism from the cyclic L ∞ -algebra (H, ω| H , S) on the cohomology H to
the original L ∞ -algebra (V , ω, L). Formulas (2.36) and (2.37) suggest one possible
definition of a morphism in a category of cyclic L ∞ -algebras as a map (in general a
nonlinear one) between the underlying vector spaces compatible with the respective
cyclic L ∞ -structures. Obviously, a map respecting symplectic structures might
seem to be a too restrictive one, the dimension of the source vector space cannot be
bigger than that of the target vector space. A possible way out would be to consider
Lagrangian correspondences instead of maps. We will not need this generalization.
Hence, we will not comment on this any further.
The minimal model defined on the subspace of physical states is then quasiisomorphic to the L ∞ -algebra defined by the original action S on the space of
all states. Formula (2.35) for the tree-level scattering amplitudes/brackets of the
minimal model is the content of the homological perturbation lemma. It expresses
the change of the trivial differential on functions on the Q-cohomology H induced
by the perturbation of the BRST operator Q by the product ∗ on functions on V .
Finally, we would like to comment on the relation to the operadic formulation
of cyclic L ∞ -algebras in Part II of this book. The maps l n or equivalently, their
duals Q
(k)
φ 0
and indeed, the classical BV action S, define a cyclic L ∞ -algebra. This
is an algebra over the cobar construction of the cyclic commutative operad briefly
recalled at the beginning of Sect. 7.2 in Part II. The minimal model construction,
when expressed by formula (2.35), applies directly to any algebra over the cobar
construction of a general cyclic operad and even more generally to any algebra over
the Feynman transform of a general modular operad. In the form presented here
it also applies to representations of cobar constructions of properads and hence, in
particular, to IBL ∞ -algebras.
Further Reading
An intuitive pedagogical account of the quantization of the relativistic point particle
can be found in:
• A. M. Polyakov, “Gauge fields and strings”, Harwood, 1987.
A standard reference for BRST and BV quantization is:
• M. Henneaux and C. Teitelboim, “Quantization of gauge systems”, Princeton
University Press, Princeton 1992.
Other useful references include:
• S. Weinberg, “The quantum theory of fields Vol. II”, Cambridge University Press,
2005,