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: <math>\Sigma E</math> is the sum of the energies of the particles : <math>\Sigma E</math> is the sum of the energies of the particles
: <math>\Sigma pc</math> is the vector sum of the ] of the particles (includes both magnitude and direction of the momenta) times the speed of light, <math>c</math> : <math>\Sigma pc</math> is the vector sum of the ] of the particles (includes both magnitude and direction of the momenta) times the speed of light, <math>c</math>

A simple way of deriving this relation is by using the momentum four-vector (in natural units):
:<math>p_i^\mu=\left(E_i,\mathbf{p}_i\right)</math>
:<math>P^\mu=\left(\Sigma E_i,\Sigma \mathbf{p}_i\right)</math>
:<math>P^\mu P_\mu=\eta_{\mu\nu}P^\mu P^\nu=(\Sigma E_i)^2-(\Sigma \mathbf{p}_i)^2=W^2</math>, since the norm of any four-vector is invariant.


== See also == == See also ==

Revision as of 02:13, 3 March 2006

In particle physics, the mathematical combination of a particle's energy and its momentum to give a value for the mass of the particle at rest. The invariant mass is the same for all frames of reference (see Special Relativity).

The invariant mass of a system of decay particles is related to the rest mass of the original particle by the following equation:

W 2 c 4 = ( Σ E ) 2 − ( Σ pc ) 2 {\displaystyle {\mbox{W}}^{2}{\mbox{c}}^{4}=(\Sigma {\mbox{E}})^{2}-(\Sigma {\mbox{pc}})^{2}}

Where:

W {\displaystyle W} is the invariant mass of the system of particles
Σ E {\displaystyle \Sigma E} is the sum of the energies of the particles
Σ p c {\displaystyle \Sigma pc} is the vector sum of the momenta of the particles (includes both magnitude and direction of the momenta) times the speed of light, c {\displaystyle c}

A simple way of deriving this relation is by using the momentum four-vector (in natural units):

p i μ = ( E i , p i ) {\displaystyle p_{i}^{\mu }=\left(E_{i},\mathbf {p} _{i}\right)}
P μ = ( Σ E i , Σ p i ) {\displaystyle P^{\mu }=\left(\Sigma E_{i},\Sigma \mathbf {p} _{i}\right)}
P μ P μ = η μ ν P μ P ν = ( Σ E i ) 2 − ( Σ p i ) 2 = W 2 {\displaystyle P^{\mu }P_{\mu }=\eta _{\mu \nu }P^{\mu }P^{\nu }=(\Sigma E_{i})^{2}-(\Sigma \mathbf {p} _{i})^{2}=W^{2}} , since the norm of any four-vector is invariant.

See also

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