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In physics, particularly in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, configurations of a physical system that satisfy classical equations of motion are called "on the mass shell" or simply more often on shell; while those that do not are called "off the mass shell", or off shell. In quantum field theory, virtual particles are termed off shell because they do not satisfy the energy–momentum relation; real exchange particles do satisfy this relation and are termed on shell (mass shell). In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle are on shell and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
s is another on-shell theorem.


Mass shell

Mass shell is a synonym for mass hyperboloid, meaning the
hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by defo ...
in energy
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
space describing the solutions to the equation: :E^2 - , \vec \,, ^2 c^2 = m_0^2 c^4, the mass–energy equivalence formula which gives the energy E in terms of the momentum \vec and the rest mass m_0 of a particle. The equation for the mass shell is also often written in terms of the four-momentum; in
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
with metric signature (+,−,−,−) and units where the speed of light c = 1, as p^\mu p_\mu \equiv p^2 = m^2. In the literature, one may also encounter p^\mu p_\mu = - m^2 if the metric signature used is (−,+,+,+). The four-momentum of an exchanged virtual particle X is q_\mu, with mass q^2 = m_X^2. The four-momentum q_\mu of the virtual particle is the difference between the four-momenta of the incoming and outgoing particles. Virtual particles corresponding to internal
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. In ...
s in a Feynman diagram are in general allowed to be off shell, but the amplitude for the process will diminish depending on how far off shell they are. This is because the q^2-dependence of the propagator is determined by the four-momenta of the incoming and outgoing particles. The propagator typically has singularities on the mass shell.Thomson, M. (2013). ''Modern particle physics''. Cambridge University Press, , p.119. When speaking of the propagator, negative values for E that satisfy the equation are thought of as being on shell, though the classical theory does not allow negative values for the energy of a particle. This is because the propagator incorporates into one expression the cases in which the particle carries energy in one direction, and in which its antiparticle carries energy in the other direction; negative and positive on-shell E then simply represent opposing flows of positive energy.


Scalar field

An example comes from considering a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
in ''D''-dimensional Minkowski space. Consider a Lagrangian density given by \mathcal(\phi,\partial_\mu \phi). The action :S = \int d^D x \mathcal(\phi,\partial_\mu \phi) The Euler–Lagrange equation for this action can be found by varying the field and its derivative and setting the variation to zero, and is: :\partial_\mu \frac = \frac Now, consider an infinitesimal spacetime translation x^\mu \rightarrow x^\mu +\alpha^\mu. The Lagrangian density \mathcal is a scalar, and so will infinitesimally transform as \delta \mathcal = \alpha^\mu \partial_\mu \mathcal under the infinitesimal transformation. On the other hand, by Taylor expansion, we have in general :\delta \mathcal = \frac \delta \phi + \frac \delta( \partial_\mu \phi) Substituting for \delta \mathcal and noting that \delta( \partial_\mu \phi) = \partial_\mu ( \delta \phi) (since the variations are independent at each point in spacetime): :\alpha^\mu \partial_\mu \mathcal = \frac \alpha^\mu \partial_\mu \phi + \frac \alpha^\mu \partial_\mu \partial_\nu \phi Since this has to hold for independent translations \alpha^\mu = (\epsilon, 0,...,0) , (0,\epsilon, ...,0), ..., we may "divide" by \alpha^\mu and write: : \partial_\mu \mathcal = \frac \partial_\mu \phi + \frac \partial_\mu \partial_\nu \phi This is an example of equation that holds ''off shell'', since it is true for any fields configuration regardless of whether it respects the equations of motion (in this case, the Euler–Lagrange equation given above). However, we can derive an ''on shell'' equation by simply substituting the Euler–Lagrange equation: : \partial_\mu \mathcal = \partial_\nu \frac \partial_\mu \phi + \frac \partial_\mu \partial_\nu \phi We can write this as: : \partial_\nu \left (\frac \partial_\mu \phi -\delta^\nu_\mu \mathcal \right) = 0 And if we define the quantity in parentheses as T^\nu_\mu, we have: :\partial_\nu T^\nu_\mu = 0 This is an instance of Noether's theorem. Here, the conserved quantity is the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
, which is only conserved on shell, that is, if the equations of motion are satisfied.


References

{{DEFAULTSORT:On Shell And Off Shell Quantum field theory