St-Petersburg State University Research Institute of Physics Faculty of Physics High Energy and Elementary Particles Physics
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There are three different formulations of Quantum mechanics:
canonical quantization (or geometrical quantization), star product
quantization and path integral quantization. It is well known that the
first one manifests the ordering ambiguity: to properly define the
model it is not sufficient to produce a Hamiltonian, also one must fix
the so-called normal ordering, the way to pass from coordinates at the
phase space to the operators. Different choice of ordering can be
compensated by quantum corrections to the Hamiltonian, therefore a
kind of "gauge symmetry" present. This symmetry also appears when one
uses the star-product approach. However, in the path integral approach
it is hidden.

Our suggestion is that this ambiguity appears in the definition of the
classical action. Usually the Lagrangian is a (polynomial) function of
coordinates and velocities. But functions we integrate on are
non-differentiable (w.r.t. time). Therefore we need to define a proper
extension of the classical action. It can be done with the help of
stochastic calculus and is not unique - this is here where the
ordering ambiguity appears.

We consider several examples which come from financial word: the
stochastic calculus is conventional tool to study the financial market
properties. We translate this toolkit to the path integral language
and will see in details how ordering ambiguity (known in the
stochastic calculus community as Ito and Stratonovich integrals)
appears. Also we will see why different ordering schemes are natural
for Quantum Mechanics and Probability Theory (Wick rotation is not,
therefore, enough to pass from quantum mechanical to thermodynamical
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March 2009