In theoretical physics

Theoretical physics

Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

, the Mandelstam variables are numerical quantities that encode the energy

Energy

In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

, momentum

Momentum

In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

, and angles of particles in a scattering process in a Lorentz-invariant fashion. They are used for scattering processes of two particles to two particles.

If the Minkowski Metric is chosen to be , the Mandelstam variables are then defined by

• 
• 
• 

Where p₁ and p₂ are the four-momenta

Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime...

 of the incoming particles and p₃ and p₄ are the four-momenta of the outgoing particles, and we are using Planck units

Planck units

In physics, Planck units are physical units of measurement defined exclusively in terms of five universal physical constants listed below, in such a manner that these five physical constants take on the numerical value of 1 when expressed in terms of these units. Planck units elegantly simplify...

 (c=1).

s is also known as the square of the center-of-mass energy (invariant mass

Invariant mass

The invariant mass, rest mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations...

) and t is also known as the square of the momentum transfer.

Feynman diagrams

The letters are also used in the terms s-channel, t-channel, u-channel. These channels represent different Feynman diagram

Feynman diagram

Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...

s or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals , respectively.

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For example the s-channel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: the s-channel is the only way that resonances and new unstable particles may be discovered provided their lifetimes are long enough that they are directly detectable. The t-channel represents the process in which the particle 1 emits the intermediate particle and becomes the final particle 3, while the particle 2 absorbs the intermediate particle and becomes 4. The u-channel is the t-channel with the role of the particles 3,4 interchanged.

The Mandelstam variables were first introduced by physicist Stanley Mandelstam

Stanley Mandelstam

Stanley Mandelstam is a South African-born theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations...

 in 1958.

High-energy limit

In the relativistic limit rest mass can be neglected, so for example,

because and It is reminded that by relativistic limit one means that the momentum (speed) is so large that in the relativistic energy-momentum equation the energy becomes essentially the momentum norm (e.g. ).

In summary,
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Addition of

Note that
where is the mass of particle .

Proof

To prove this, we need to use two facts:

• The square of a particle's four momentum is the square of its mass,

• And conservation of four-momentum,

So, to begin,

First, use (1) to re-write these,

Then add them
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Then use eq (2) to simplify further,
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So finally,

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See also

• Feynman diagrams
• Bhabha scattering
  Bhabha scattering
  In quantum electrodynamics, Bhabha scattering is the electron-positron scattering process:There are two leading-order Feynman diagrams contributing to this interaction: an annihilation process and a scattering process. The Bhabha scattering rate is used as a luminosity monitor in electron-positron...
  
  
• Møller scattering
  Møller scattering
  Møller scattering is the name given to electron-electron scattering in Quantum Field Theory. The electron interaction that is idealized in Møller scattering forms the theoretical basis of many familiar phenomena such as the repulsion of electrons in the Helium nucleus...
  
  
• Compton scattering
  Compton scattering
  In physics, Compton scattering is a type of scattering that X-rays and gamma rays undergo in matter. The inelastic scattering of photons in matter results in a decrease in energy of an X-ray or gamma ray photon, called the Compton effect...