On shell and off shell

 ( Quantum Field Theory ) 

In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on the mass shell ( on shell); while those that do not are called off the mass shell ( off shell).

In quantum field theory, are termed off shell because they do not satisfy the energy–momentum relation; real exchange particles do satisfy this relation and are termed on (mass) shell.Thomson, M. (2013). Modern particle physics. Cambridge University Press, , pp. 117–119. 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 is another on-shell theorem.

---

Mass shell Mass shell is a synonym for mass hyperboloid, meaning the hyperboloid in energy–momentum space describing the solutions to the equation:

$E^2\; -\; |\backslash vec\{p\}\; \backslash ,|^2\; c^2\; =\; m\_0^2\; c^4$,

the mass–energy equivalence formula which gives the energy $E$ in terms of the momentum $\backslash vec\{p\}$ 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 with metric signature (+,−,−,−) and units where the speed of light $c\; =\; 1$, as $p^\backslash mu\; p\_\backslash mu\; \backslash equiv\; p^2\; =\; m\_0^2$. In the literature, one may also encounter $p^\backslash mu\; p\_\backslash mu\; =\; -\; m\_0^2$ if the metric signature used is (−,+,+,+).

The four-momentum of an exchanged virtual particle $X$ is $q\_\backslash mu$, with mass $q^2\; =\; m\_X^2$. The four-momentum $q\_\backslash mu$ of the virtual particle is the difference between the four-momenta of the incoming and outgoing particles.

Virtual particles corresponding to internal 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 D-dimensional Minkowski space. Consider a Lagrangian density given by $\backslash mathcal\{L\}(\backslash phi,\backslash partial\_\backslash mu\; \backslash phi)$. The action is

$S\; =\; \backslash int\; d^D\; x\; \backslash mathcal\{L\}(\backslash phi,\backslash partial\_\backslash mu\; \backslash phi)$.

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:

$\backslash partial\_\backslash mu\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash mu\; \backslash phi)\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; \backslash phi\}$

Now, consider an infinitesimal spacetime translation $x^\backslash mu\; \backslash rightarrow\; x^\backslash mu\; +\backslash alpha^\backslash mu$. The Lagrangian density $\backslash mathcal\{L\}$ is a scalar, and so will infinitesimally transform as $\backslash delta\; \backslash mathcal\{L\}\; =\; \backslash alpha^\backslash mu\; \backslash partial\_\backslash mu\; \backslash mathcal\{L\}$ under the infinitesimal transformation. On the other hand, by Taylor expansion, we have in general

$\backslash delta\; \backslash mathcal\{L\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; \backslash phi\}\; \backslash delta\; \backslash phi\; +\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash mu\; \backslash phi)\}\; \backslash delta(\; \backslash partial\_\backslash mu\; \backslash phi)$

Substituting for $\backslash delta\; \backslash mathcal\{L\}$ and noting that $\backslash delta(\; \backslash partial\_\backslash mu\; \backslash phi)\; =\; \backslash partial\_\backslash mu\; (\; \backslash delta\; \backslash phi)$ (since the variations are independent at each point in spacetime):

$\backslash alpha^\backslash mu\; \backslash partial\_\backslash mu\; \backslash mathcal\{L\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; \backslash phi\}\; \backslash alpha^\backslash mu\; \backslash partial\_\backslash mu\; \backslash phi\; +\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash nu\; \backslash phi)\}\; \backslash alpha^\backslash mu\; \backslash partial\_\backslash mu\; \backslash partial\_\backslash nu\; \backslash phi$

Since this has to hold for independent translations $\backslash alpha^\backslash mu\; =\; (\backslash epsilon,\; 0,...,0)\; ,\; (0,\backslash epsilon,\; ...,0),\; ...$, we may "divide" by $\backslash alpha^\backslash mu$ and write:

$\backslash partial\_\backslash mu\; \backslash mathcal\{L\}\; =\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; \backslash phi\}\; \backslash partial\_\backslash mu\; \backslash phi\; +\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash nu\; \backslash phi)\}\; \backslash partial\_\backslash mu\; \backslash partial\_\backslash nu\; \backslash phi$

This is an example of an 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:

$\backslash partial\_\backslash mu\; \backslash mathcal\{L\}\; =\; \backslash partial\_\backslash nu\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash nu\; \backslash phi)\}\; \backslash partial\_\backslash mu\; \backslash phi\; +\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash nu\; \backslash phi)\}\; \backslash partial\_\backslash mu\; \backslash partial\_\backslash nu\; \backslash phi$

We can write this as:

$\backslash partial\_\backslash nu\; \backslash left\; (\backslash frac\{\backslash partial\; \backslash mathcal\{L\}\}\{\backslash partial\; (\backslash partial\_\backslash nu\; \backslash phi)\}\; \backslash partial\_\backslash mu\; \backslash phi\; -\backslash delta^\backslash nu\_\backslash mu\; \backslash mathcal\{L\}\; \backslash right)\; =\; 0$

And if we define the quantity in parentheses as $T^\backslash nu\{\}\_\backslash mu$, we have:

$\backslash partial\_\backslash nu\; T^\backslash nu\{\}\_\backslash mu\; =\; 0$

This is an instance of Noether's theorem. Here, the conserved quantity is the stress–energy tensor, which is only conserved on shell, that is, if the equations of motion are satisfied.

Post Comment