In the laboratory frame, denoted \(\Sigma\), a particle travelling in the z-direction has momentum \(\mathbf{p}=p_{z} \hat{\mathbf{z}}\) and energy \(E\). (a) Use the Lorentz transformation to find expressions for the momentum \(p_{z}^{\prime}\) and energy \(E^{\prime}\) of the particle in a frame \(\Sigma^{\prime}\), which is moving in a velocity \(\mathbf{v}=+v \hat{z}\) relative to \(\Sigma\), and show that \(E^{2}-p_{2}^{2}=\left(E^{\prime}\right)^{2}-\left(p_{2}^{\prime}\right)^{2}\). (b) For a system of particles, prove that the total four-momentum squared, $$ p^{\mu} p_{\mu} \equiv\left(\sum_{i} E_{i}\right)^{2}-\left(\sum_{i} \mathbf{p}_{i}\right)^{2} $$ is invariant under Lorentz transformations.

Understanding the concepts of relativistic energy and momentum is crucial in particle physics.
When particles move at speeds close to the speed of light, their energy and momentum are described by special relativity.
In this context, the energy (E) of a particle and its momentum (\textbf{p}) are connected through the famous equations from Einstein's theory of relativity.
These equations differ significantly from the classical definitions of energy and momentum.

• Relativistic momentum in the z direction is given by: \(p_z = \gamma mv_z\), where \(m\)is the mass of the particle, \(v_z\)is the velocity in the z direction, and \(\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\).
• Relativistic energy is given by: \(E = \gamma mc^2\), where \(c\)is the speed of light.

Both quantities transform differently under changes from one frame of reference to another, as described by Lorentz transformations.
These transformations ensure the laws of physics, including the conservation of energy and momentum, hold true in all inertial frames.

Lorentz invariance is a foundational principle in the theory of relativity.
It states that the laws of physics remain the same for all observers, regardless of their constant, uniform motion relative to each other.
In mathematical terms, Lorentz invariance means that certain quantities, particularly those involving space and time coordinates, remain unchanged (invariant) when transformed between different inertial frames.
In the context of particle physics, one key Lorentz invariant quantity is the combination of a particle's energy and momentum: \(E^2 - (p_z c)^2\).
This invariant helps us prove that physical phenomena are consistent across different reference frames.
For example, if we transform the energy and momentum of a particle to a new frame using the Lorentz transformation:

• \(E' = \frac{E - vp_z}{\sqrt{1-\frac{v^2}{c^2}}}\)
• \(p_z' = \frac{p_z - \frac{vE}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}\)

We can show that: \({E'}^2 - (p_z' c)^2\) equals \(E^2 - (p_z c)^2\), proving the invariance of this quantity.
This invariance is fundamental to our understanding of space and time, ensuring that physical laws do not depend on the observer's frame of reference.

In relativity, the concept of momentum is extended to four-momentum, a vector that includes both energy and momentum as components.
The four-momentum \(p^\mu\)of a particle is written as:

• \(p^\mu = (E/c, p_x, p_y, p_z)\), combining the energy and three-dimensional momentum in a single entity.

This vector is crucial because it transforms in a simple way between different reference frames under Lorentz transformations.
The invariant magnitude of the four-momentum is given by \(p^\mu p_\mu\) or:
\(\left(\frac{E}{c}\right)^2 - \left(p_x^2 + p_y^2 + p_z^2\right)\),
which is conserved in all inertial frames.
This invariant magnitude helps us understand interactions between particles and systems of particles.
For a system of particles, the total four-momentum is the sum of the individual four-momenta:\(p^{\mu}_{\text{total}} = \sum_i p_i^{\mu}\).
As a result, the total invariant magnitude is:
\(\left(\sum_i E_i\right)^2 - \left(\sum_i \mathbf{p}_i\right)^2\).
Since each individual four-momentum is Lorentz invariant, the total four-momentum also remains invariant under Lorentz transformations.
This helps to ensure that the total energy and momentum of a closed system are conserved, reinforcing the consistency of physical laws across different reference frames.