3 Symmetry Transformations and Conservation Laws

Ashok Goyal

1. Introduction:

Principle of symmetry transformations plays a key role in the formulation of Quantum Field theories in many branches of physics like particle physics, condensed matter physics, general theory of relativity, string theories etc. Conservation laws in physics have a deep relationship with symmetry operations. The requirement of invariance of the Lagrangian under symmetry transformations so that the physical results remain unchanged leads to conservation laws. In what fallows we will consider some space time transformations leading to corresponding conservation laws. Suppose we have a Lagrangian of the system   where  ’s stand for the set generalized coordinates. If we now make a transformation of the coordinate from  , the Lagrangian in new coordinates is expressed as  . The different solutions   obtained from   and   must have equivalent trajectories (path). The coordinates transformations can be discrete or continuous. An example of a discrete transformation is ‘mirror reflection’ under which

A transformation is continuous if the transformed coordinates have a continuous dependence on a suitable parameter defining the transformation. An example is rotational transformation about the z-axis under which

2. Space time Translational symmetry and conservation of Energy and Linear Momentum.

The space is assumed to be homogeneous and isotropic. A physical system at one place in space and time developes the same way as at any other point. The laws of physics are invariant under the space-time translation. Let us consider a Lagrangian of a dynamical system of n particles. The Lagrangian is a function of position coordinates and velocities. Assuming the constraints to be holonomic, the Lagrangian can be written as

Where the potential does not depend on the velocities.

1) Symmetry under space translation:

Consider an infinitesimal translation of the coordinate

If the Lagrangian remains invariant so that the physical results remain unchanged, we have . Since  is an arbitrary displacement

and from Lagrange’s equation  of motion

Now

From (3.8)

Total linear momentum of the system is conserved.   IF the motion of the system is described by a set of

generalized coordinates   we define the quantity

as ‘generalized momenta’ or ‘conjugate momenta. In general the conjugate momenta may not be identical to the mechanical momenta and may have dimensions different from MLT-1 . The dimension of mechanical momenta. In terms of generalized force   the   Lagrangian equation

If in the expression of the Lagrangian, a particular coordinate   or ‘ignorable’ and obviously for a cyclic coordinate does not appear explicitly, it is called ‘cyclic’

Hence the momentum conjugate to acyclic coordinate is conserved. ii) Symmetry under time translation. Under time translation

The Lagrangian is transformed to 

where

If the Lagrangian remains invariant under time translation

Where we have used the Lagrange equation in the first term on the right to express  

which implies that the total energy is conserved.

3. Rotational symmetry

Let a vector   rotates around the z-axis and rotates to  after rotating by an angle   as shown.  The tip of the vector  moves along a circle of radius .

Langrangi equation is given by

Total angular momentum is conserved.

Invariance of the Lagrangian under rotation is justified because of the isotropy of space.

4. Neother’s Theorem

Suppose we have a Lagrangian which depends on the generalized coordinates, generalized velocities and time and a continuous symmetry operation denoted by  which depends on the parameter set  and .For example, the symmetry operation of rotation about the Z-axis is parameterized by a continuous angle of rotation .  The transformed Lagrangian  is a function of  

If  is a symmetry operation of the system, we must have

Since eqn. (3.30) holds for all s.

This is Noether’s Theorem.

Noether’s Theorem states that if a Lagrangian is invariant under a continuous symmetry operation parameterized by M parameters , then there are M conserved quantities associated with the symmetry transformation.

This theorem formed the corner stone in the development of Quantum Field Theory.

Example: Consider a system with the Lagrangian

 are the generalized coordinates and the potential depends only on T. The Lagrangian does not involve the coordinates  and , therefore they are cyclic coordinates. We then have the momenta conjugate to  and . Conserved.

5. Summary:

Homogeneity and Isotropy of space leads to the conservation of energy linear momentum and angular momentum shown in the table.

Symmetry	Lagrangian	Conserved quantity
1. Space translation Homogeneity	Invariant under space transition	Linear momentum
2.Time translation homogeneity of time	Invariant under translation in time and no explicit dependence on time	Total Energy
3. Rotation Isotropy of space	Invariant under rotation	Angular momentum

There are seven constraints (integrals of motion) for a closed system; total energy, three components of linear momentum and three components of angular momentum.

• If a Lagrangian is invariant under a continuous transformation parameterized by M parameters, then according to Noether’s theorem, there are M conserved quantities associated with these symmetry transformations.