Noether’s Theorem
Every continuous symmetry in a physical system corresponds to a conservation law.
In simpler terms, whenever nature exhibits a symmetry, there exists a physical quantity that remains unchanged.
What exactly is a symmetry?
A symmetry is a transformation that leaves the fundamental laws of a system unaffected.
If a physical system behaves identically before and after such a transformation, we say it is invariant under that symmetry.
Symmetries can appear in space or time - such as translations and rotations - or concern the internal structure of fields, as in the case of gauge symmetries.
Noether's theorem forms the mathematical bridge linking symmetries to the conservation laws of physics.
Here are a few key examples:
| Symmetry | Conservation Law | Conserved Quantity |
|---|---|---|
| Time invariance (the laws of physics do not change over time) | Conservation of energy | Total energy |
| Spatial invariance (the laws are the same everywhere in space) | Conservation of momentum | Linear momentum |
| Rotational invariance (the laws are independent of orientation) | Conservation of angular momentum | Angular momentum |
| Gauge invariance (the laws remain unchanged under a change in the fields' "internal phase") | Conservation of electric charge | Electric charge |
Noether's theorem is far more than a mathematical curiosity - it is the cornerstone of modern physics. It reveals that conservation laws are not separate assumptions, but rather direct consequences of the fundamental symmetries of the universe.
Note. Albert Einstein himself acknowledged that without Emmy Noether's insight (1917), general relativity would not have achieved a consistent mathematical foundation.
Mathematical Formulation
The theorem applies to systems described by a Lagrangian L that depends on the generalized coordinates $ q_i $, their time derivatives $ \frac{dq_i}{dt} $, and the time variable $ t $:
$$ L = L(q_i, \dot{q}_i, t) $$
If the Lagrangian remains unchanged under a continuous transformation of $ q_i $ - or changes only by a total time derivative - then a conserved quantity must exist.
$$ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} , \delta q_i \right) = 0 $$
The term in parentheses is known as the Noether charge - the physical quantity that remains constant in time.
Example 1: Time Invariance
If the Lagrangian does not depend explicitly on time, the corresponding symmetry is a translation in time:
$$ \frac{ \partial L}{\partial t} = 0 $$
This means that if we shift the entire system in time - from $ t $ to $ t + \Delta t $ - the form of L remains unchanged.
In other words, moving everything forward by a time interval $ \Delta t $ leaves the equations of motion identical: the laws of physics are the same today, tomorrow, and a thousand years from now.
This is called a temporal translation symmetry.
In this case, the total energy of the system is conserved.
Note. A physical system is described by a Lagrangian: $$ L(q_i, \dot{q}_i, t) $$ Its equations of motion arise from the principle of least action: $$ \delta S = 0 \quad \text{where} \quad S = \int_{t_1}^{t_2} L(q_i, \dot{q}_i, t)\, dt $$ From this principle we derive the Euler - Lagrange equations: $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 $$ Now, if the Lagrangian does not depend explicitly on time, that is: $$ \frac{\partial L}{\partial t} = 0 $$ then translating the system in time leaves $ L $ unchanged. In other words, the equations of motion are identical - a manifestation of temporal symmetry. According to Noether's theorem, each continuous symmetry corresponds to a conserved quantity. For time symmetry, that conserved quantity is the system's total energy: $$ E = \sum_i \dot{q}_i \frac{\partial L}{\partial \dot{q}_i} - L $$ Taking the time derivative of $ E $ gives: $$ \frac{dE}{dt} =
\frac{d}{dt}\left(\sum_i \dot{q}_i \frac{\partial L}{\partial \dot{q}_i} - L\right) $$ Expanding this expression: $$ \frac{dE}{dt} =
\sum_i \ddot{q}_i \frac{\partial L}{\partial \dot{q}_i} + \sum_i \dot{q}_i \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)
- \sum_i \frac{\partial L}{\partial q_i} \dot{q}_i - \frac{\partial L}{\partial t} $$ From the Euler - Lagrange equations, $$
\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) = \frac{\partial L}{\partial q_i} $$ so the middle terms cancel, leaving: $$ \frac{dE}{dt} = -\frac{\partial L}{\partial t} $$ If $ L $ does not depend explicitly on time, i.e. $ \partial L / \partial t = 0 $, then: $$ \frac{dE}{dt} = 0 $$ The total energy of the system is therefore conserved.
Example 2: Spatial Invariance
If the Lagrangian does not depend on the absolute position $ x $, but only on relative distances, the symmetry corresponds to a spatial translation.
In this case, the conserved quantity is the linear momentum.
Example 3: Rotational Invariance
If the Lagrangian remains unchanged when the system is rotated, the symmetry is rotational.
The corresponding conserved quantity is the angular momentum.
And so forth - each symmetry in nature gives rise to a conservation law through Noether's remarkable insight.
