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Classical mechanics is based on conservation laws which represent the symmetries of spacetime. The lagrangian function L is a function of position and velocity while the hamiltonian is a function of position and momentum. The velocity and momentum descriptions are related by a legendre transformation:

H = pv - L

which is the integration by parts formula ∫ p dv = pv - ∫ v dp. The lagrangian description is in the tangent bundle and the hamiltonian description is in the cotangent bundle. The structure of the cotangent bundle is encoded in the tautological one-form θ = p dx. The symplectic 2-form dθ is dp ∧ dx. Conservation laws are of the form d*j = 0 where j is the conserved current. Momentum is the generator and conserved quantity of spacetime translations exp(ix'p)f(x) = f(x + x'). The forces are generated by potentials F = -dU. The lagrangian is the sum of the kinetic energy T and potential energy U:

L = T + U = 1/2 mv

The hamiltonian is the sum of the kinetic and potential energies, H = T + U. Symmetries of spacetime are represented by killing vector fields. The conserved currents are T

The hamiltonian description is based on the phase space which is the space of (x, p). This phase space is the space of motions or the space of solutions to the equations of motion. This space can be described abstractly by the symplectic 2-form ##d\alpha \wedge d\tilde{\alpha}##. Take a field

##\phi(x) = \sum_k \frac{1}{\sqrt{2E}} \alpha(k) e^{ipx} + \tilde{\alpha}(k) e^{-ipx}##

the phase space description can be made covariant by taking the paths or solutions the be the states, rather than the initial conditions. Take a path ϒ

## \gamma \rightarrow \gamma + \epsilon \frac{\partial \tilde{\alpha}}{\partial \epsilon} \frac{\partial}{\partial \tilde{\alpha}} ##

where ##\alpha = \frac{\delta}{\delta \tilde{\alpha}}##

##\alpha## is a Heisenberg picture operator which evolves as ##\alpha \rightarrow e^{itE} \alpha e^{-itE} ##. The alphas satify the commutation relation [α, α] = 1. ## [\alpha, \tilde{\alpha}] = 1##. A path is a point in phase space. The usual notion of evolution in time f(t) → f(t + x) must be interpreted not as the value of f at a later time t, but a function f translated in time by x. what is the correct way to describe the operator formalism in a covariant way? In my next post I will describe details. The energy momentum T generates conformal transformations. In string theory, the states of particles are always on-shell.

H = pv - L

which is the integration by parts formula ∫ p dv = pv - ∫ v dp. The lagrangian description is in the tangent bundle and the hamiltonian description is in the cotangent bundle. The structure of the cotangent bundle is encoded in the tautological one-form θ = p dx. The symplectic 2-form dθ is dp ∧ dx. Conservation laws are of the form d*j = 0 where j is the conserved current. Momentum is the generator and conserved quantity of spacetime translations exp(ix'p)f(x) = f(x + x'). The forces are generated by potentials F = -dU. The lagrangian is the sum of the kinetic energy T and potential energy U:

L = T + U = 1/2 mv

^{2}+ U(x)The hamiltonian is the sum of the kinetic and potential energies, H = T + U. Symmetries of spacetime are represented by killing vector fields. The conserved currents are T

^{ij}k_{j}where k are the killing vectors. T generates conformal transformations. The equations of motion are d*T = 0. In the hamiltonain description, the equations of motion are found from the Lie brackets, {H, X} = dX/dt = dH/dp and {H, P} = dP/dt = -dH/dx. The lagrangian equations of motion are found from δS = 0. Because of the difference between time and space, the relationship between energy and momentum is not exactly symmetric. Momentum is the flux of energy. This can be proved from the fact that T is a symmetric tensor. The conservation laws are closely related to the fact dd = 0, which is due to the fact that the boundary of a boundary is zero.The hamiltonian description is based on the phase space which is the space of (x, p). This phase space is the space of motions or the space of solutions to the equations of motion. This space can be described abstractly by the symplectic 2-form ##d\alpha \wedge d\tilde{\alpha}##. Take a field

##\phi(x) = \sum_k \frac{1}{\sqrt{2E}} \alpha(k) e^{ipx} + \tilde{\alpha}(k) e^{-ipx}##

the phase space description can be made covariant by taking the paths or solutions the be the states, rather than the initial conditions. Take a path ϒ

## \gamma \rightarrow \gamma + \epsilon \frac{\partial \tilde{\alpha}}{\partial \epsilon} \frac{\partial}{\partial \tilde{\alpha}} ##

where ##\alpha = \frac{\delta}{\delta \tilde{\alpha}}##

##\alpha## is a Heisenberg picture operator which evolves as ##\alpha \rightarrow e^{itE} \alpha e^{-itE} ##. The alphas satify the commutation relation [α, α] = 1. ## [\alpha, \tilde{\alpha}] = 1##. A path is a point in phase space. The usual notion of evolution in time f(t) → f(t + x) must be interpreted not as the value of f at a later time t, but a function f translated in time by x. what is the correct way to describe the operator formalism in a covariant way? In my next post I will describe details. The energy momentum T generates conformal transformations. In string theory, the states of particles are always on-shell.

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