LorentzAndPoincaréInvariance

The concepts of Lorentz invariance and Poincaré invariance come from the study of spacetime symmetries in the theory of special relativity and quantum field theory. They're mathematical frameworks that describe how physical laws remain unchanged under certain transformations. Here's a detailed comparison:

Lorentz Invariance:

The Lorentz transformation is a linear transformation that relates the coordinates of an event in one inertial frame to the coordinates of the same event in another inertial frame. These transformations leave the spacetime interval invariant. Mathematically, if we have two events separated by a space-time interval $(\Delta x, \Delta y, \Delta z, \Delta t)$, a Lorentz transformation in the x direction will look like:

$$\Delta x' = \gamma(\Delta x - v\Delta t)$$

$$\Delta y' = \Delta y$$

$$\Delta z' = \Delta z$$

$$\Delta t' = \gamma(\Delta t - \frac{v\Delta x}{c^2})$$

where $v$ is the relative velocity between the two frames, $c$ is the speed of light, and $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is the Lorentz factor.

Lorentz invariance then refers to the property of a physical law that it remains the same under Lorentz transformations. A physical law is said to be Lorentz invariant if its mathematical description does not change when transformed under Lorentz transformations. This is a foundational principle in special relativity.

Poincaré Invariance:

The Poincaré group is a group of transformations that include both the Lorentz transformations and translations in space and time. A translation in space and time is a shift of the origin of the coordinate system, without any rotation or reflection of the axes.

Mathematically, if we have a space-time point $(x, y, z, t)$, a translation will look like:

$$x' = x + a$$

$$y' = y + b$$

$$z' = z + c$$

$$t' = t + d$$

where $a, b, c, d$ are constants representing shifts along each axis.

Poincaré invariance then refers to the property of a physical law that it remains the same under Poincaré transformations (both Lorentz transformations and translations). A physical law is said to be Poincaré invariant if its mathematical description does not change when transformed under Poincaré transformations. This is a foundational principle in quantum field theory.

Difference:

The main difference between Lorentz invariance and Poincaré invariance is that the former only considers transformations related to rotation and boost (change of velocity), preserving the origin of the coordinate system, while the latter also includes translations, shifts of the origin in space and time.

This means Poincaré invariance is a more general symmetry that includes Lorentz invariance. All Lorentz invariant laws are Poincaré invariant, but not all Poincaré invariant laws are necessarily Lorentz invariant if they're not invariant under spatial or temporal translation. The more general Poincaré invariance is especially important in quantum field theory, where translation invariance leads to the important concept of momentum conservation.