Analysis of flows
Encyclopedia
In theoretical physics
, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" (i.e. flows the formulation of a theory is invariant under). It is generally agreed that flows indicate nothing more than a redundancy in the description of the dynamics of a system, but often, it is simpler computationally to work with a redundant description.
. The on-shell solutions are given by the variational problem of extremizing the action subject to boundary conditions.
While the boundary is often ignored in textbooks, it is crucial in the study of flows. Suppose we have a "flow", i.e. the generator
of a smooth one-dimensional group of transformations of the configuration space, which maps on-shell states to on-shell states while preserving the boundary conditions. Because of the variational principle, the action for all of the configurations on the orbit is the same. This is not the case for more general transformations which map on shell to on shell states but change the boundary conditions.
Here are several examples. In a theory with translational symmetry
, timelike translations are not flows because in general they change the boundary conditions. However, now take the case of a simple harmonic oscillator, where the boundary points are at a separation of a multiple of the period from each other, and the initial and final positions are the same at the boundary points. For this particular example, it turns out there is a flow. Even though this is technically a flow, this would usually not be considered a gauge symmetry because it is not local.
Flows can be given as derivations
over the algebra of smooth functionals over the configuration space. If we have a flow distribution (i.e. flow-valued distribution) such that the flow convolved over a local region only affects the field configuration in that region, we call the flow distribution a gauge flow.
Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals/flows which agree on shell.
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, an analysis of flows is the study of "gauge" or "gaugelike" "symmetries" (i.e. flows the formulation of a theory is invariant under). It is generally agreed that flows indicate nothing more than a redundancy in the description of the dynamics of a system, but often, it is simpler computationally to work with a redundant description.
Flows in the action formalism
Classically, the action is a functional on the configuration spaceConfiguration space
- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...
. The on-shell solutions are given by the variational problem of extremizing the action subject to boundary conditions.
While the boundary is often ignored in textbooks, it is crucial in the study of flows. Suppose we have a "flow", i.e. the generator
Generating set
In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:...
of a smooth one-dimensional group of transformations of the configuration space, which maps on-shell states to on-shell states while preserving the boundary conditions. Because of the variational principle, the action for all of the configurations on the orbit is the same. This is not the case for more general transformations which map on shell to on shell states but change the boundary conditions.
Here are several examples. In a theory with translational symmetry
Translational symmetry
In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...
, timelike translations are not flows because in general they change the boundary conditions. However, now take the case of a simple harmonic oscillator, where the boundary points are at a separation of a multiple of the period from each other, and the initial and final positions are the same at the boundary points. For this particular example, it turns out there is a flow. Even though this is technically a flow, this would usually not be considered a gauge symmetry because it is not local.
Flows can be given as derivations
Differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with a derivation, which is a unary function that is linear and satisfies the Leibniz product law...
over the algebra of smooth functionals over the configuration space. If we have a flow distribution (i.e. flow-valued distribution) such that the flow convolved over a local region only affects the field configuration in that region, we call the flow distribution a gauge flow.
Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler-Lagrange equations, or in other words, consider the equivalence class of functionals/flows which agree on shell.
Flows in the Hamiltonian formalism
- First class constraints
- Second class constraintsSecond class constraintsIn a constrained Hamiltonian system, a dynamical quantity is second class if its Poisson bracket with at least one constraint is nonvanishing...
- BRST formalism
- Batalin-Vilkovisky