Symmetries in Physics
In physics, a symmetry is a transformation that leaves the fundamental laws of nature unchanged.
Every symmetry expresses an invariance: something that remains constant even when a transformation is applied.
According to Noether’s theorem, each continuous symmetry corresponds to a conserved quantity:
- Time symmetry
If the laws of physics are the same at all times, energy is conserved. - Spatial symmetry
If the laws of physics are the same everywhere in space, linear momentum is conserved. - Rotational symmetry
If the laws of physics do not depend on the orientation of the system, angular momentum is conserved.
For example, rotating a mechanical system in space does not change the equations that describe its motion. That is what we call rotational symmetry.
Other examples. Moving an experiment forward or backward in time produces the same results, which shows time symmetry. Exchanging two identical particles without changing the overall configuration represents a permutation symmetry. Many physical laws share this kind of hidden structure.
Groups: the mathematical language of symmetry
Symmetries can be combined. Performing one transformation after another is equivalent to applying a single, combined transformation.
All such transformations together form a mathematical structure called a group.
In abstract algebra, a group is a set of elements and an operation that satisfies four key rules:
- Closure: combining any two transformations gives another valid transformation of the same type.
- Associativity: grouping transformations in a different order does not affect the outcome.
- Identity element: there exists one transformation that changes nothing (the identity).
- Inverse element: every transformation can be undone by another.
For instance, all possible rotations in space form the group SO(3), short for “Special Orthogonal.” It consists of orthogonal matrices with determinant equal to 1.
In particle physics, internal symmetries rely on groups like SU(2) and SU(3), where SU means “Special Unitary.” These groups consist of unitary matrices with determinant equal to 1.
Further examples. Translations in space and time form an abelian group. Relativistic rotations and boosts form the Lorentz group. Each of these captures a different kind of symmetry in nature.
Physicists often describe these symmetries using matrices. Matrices provide a convenient way to express transformations as linear operations acting on vectors.
For example, a rotation in three-dimensional space can be represented by a 3×3 orthogonal matrix acting on a vector.
More examples. Lorentz transformations are represented by 4×4 matrices acting on space-time four-vectors. Symmetries like isospin or color charge in particle physics use unitary matrices belonging to SU(2) or SU(3) groups.
These matrices obey the same combination rules as the underlying group: multiplying two matrices corresponds to applying two transformations in sequence.
This is why physicists often refer to them as matrix groups.
| Physical concept | Mathematical structure | Representation |
|---|---|---|
| Symmetry of a physical law | Group of transformations | Matrices acting on vectors |
| Invariance (what remains unchanged) | Property of the group | Conserved quantity (Noether) |
| Physical transformation (rotation, inversion, etc.) | Element of the group | Corresponding matrix |
A concrete example
Let’s take a simple case: a free particle moving in three-dimensional space.
Its motion remains the same if we rotate the coordinate system. This is a rotational symmetry.
Rotations belong to the group SO(3). Each rotation can be represented by a 3×3 orthogonal matrix acting on the particle’s position vector \( \vec{r} = (x, y, z) \).
$$
R(\hat{r}, \theta) =
\begin{pmatrix}
\cos\theta + r_x^2(1 - \cos\theta) & r_x r_y(1 - \cos\theta) - r_z \sin\theta & r_x r_z(1 - \cos\theta) + r_y \sin\theta \\[6pt]
r_y r_x(1 - \cos\theta) + r_z \sin\theta & \cos\theta + r_y^2(1 - \cos\theta) & r_y r_z(1 - \cos\theta) - r_x \sin\theta \\[6pt]
r_z r_x(1 - \cos\theta) - r_y \sin\theta & r_z r_y(1 - \cos\theta) + r_x \sin\theta & \cos\theta + r_z^2(1 - \cos\theta)
\end{pmatrix}.
$$
This symmetry is what leads to the conservation of angular momentum.
$$ R_z(90^\circ)= \begin{pmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}. $$
Suppose the particle’s initial position is \( \mathbf r=(2,\,1,\,0) \).
After applying the rotation, we get:
$$ \mathbf r' = R_z\,\mathbf r = \begin{pmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 2\\[2pt] 1\\[2pt] 0 \end{pmatrix} = \begin{pmatrix} -1\\[2pt] 2\\[2pt] 0 \end{pmatrix}. $$
The particle’s position becomes \( \mathbf r=(-1,\,2,\,0) \).
As expected, the vector’s length stays the same:
$$ \|\mathbf r\|^2 = 2^2+1^2+0^2 = 5, \qquad \|\mathbf r'\|^2 = (-1)^2+2^2+0^2 = 5. $$
Now consider the particle’s momentum \( \mathbf p=(0,\,3,\,0) \). After rotation:
$$ \mathbf p' = R_z\,\mathbf p = \begin{pmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0\\[2pt] 3\\[2pt] 0 \end{pmatrix} = \begin{pmatrix} -3\\[2pt] 0\\[2pt] 0 \end{pmatrix}. $$
The angular momentum before rotation is given by \( \mathbf L=\mathbf r\times \mathbf p \):
$$ \mathbf L = \begin{pmatrix} 2\\[2pt] 1\\[2pt] 0 \end{pmatrix} \times \begin{pmatrix} 0\\[2pt] 3\\[2pt] 0 \end{pmatrix} = \begin{pmatrix} 0\\[2pt] 0\\[2pt] 6 \end{pmatrix}. $$
After the rotation, \( \mathbf L'=\mathbf r'\times \mathbf p' \):
$$ \mathbf L' = \begin{pmatrix} -1\\[2pt] 2\\[2pt] 0 \end{pmatrix} \times \begin{pmatrix} -3\\[2pt] 0\\[2pt] 0 \end{pmatrix} = \begin{pmatrix} 0\\[2pt] 0\\[2pt] 6 \end{pmatrix}. $$
The angular momentum remains exactly the same before and after the rotation.
$$ \mathbf L = \mathbf L' = \begin{pmatrix} 0\\[2pt] 0\\[2pt] 6 \end{pmatrix} $$
This simple calculation shows two essential points. First, rotations are represented by 3×3 orthogonal matrices that preserve lengths. Second, when a system is rotationally symmetric, its angular momentum remains constant. In this example, \( \mathbf L \) does not change when both \( \mathbf r \) and \( \mathbf p \) are rotated through the same transformation in \(SO(3)\).
Symmetry is not just a mathematical curiosity: it lies at the heart of how the universe conserves energy, momentum, and other fundamental quantities. It reveals the deep unity between the structure of space-time and the laws that govern it.
