Symmetry at the Heart of Physics

Why Symmetry is What Physics is All About

It takes a fair few years of a university degree to begin to realise that symmetry is perhaps the most fundamental concept in modern physics. It’s no accident that one of the leading candidates for new physics is called “Supersymmetry”.

Anyone who studied physics at high-school has actually come across an example of symmetry’s influence in physics, but may not have realised it. Energy conservation and momentum conservation are both rules that nearly everyone is aware of, but a much smaller number of people will realise that these rules are actually consequences of symmetry.

At high-school you’re just told to accept these conservation laws without asking too many questions. If you were to ask your physics teacher why it is exactly that these laws are true, you might just receive the response that it’s an empirical fact. We’ve done experiments, and in all experiments we’ve conducted we’ve noticed that energy and momentum are always conserved, so we’ve elevated that observation into a universal law.

Like most physics, this is probably the way we first came up with these conservation laws – because we had observed them to hold in the experiments we performed. You’ll be pleased to discover however that now we know much more about energy and momentum, this is not the only evidence we have for the universality of conservation laws, we can in fact derive the conservation of energy and momentum from an alternative formulation of mechanics known as “Lagrangian Mechanics”.

Lagrangian Mechanics

At a high-school level, the way you’d be used to doing mechanics is through Newton’s Laws. But there’s a much more powerful and simpler way of doing classical mechanics by using something called the “Lagrangian”.

The Lagrangian is a function \mathcal{L} which we write as the kinetic energy minus the potential energy \mathcal{L} = KE - PE. In classical mechanics then, if we’re dealing with a free particle moving in the depths of empty space, we can write the Lagrangian as being equal to the kinetic energy of the particle

{\mathcal{L}=\frac{1}{2}mv^2}

What we then use to find the equation of motion of the system is something called the “Euler-Lagrange equation”, which looks like this:

{\frac{d}{dt}(\frac{\partial\mathcal{L}}{\partial v})=\frac{\partial \mathcal{L}}{\partial x}}

In the case for a free particle above, this gives us the equation of motion:

{m\frac{dv}{dt}=0}

which is just Newton’s Second Law of Motion!

Hopefully this example above gives you some evidence for believing my claim that Newtonian Mechanics can be rewritten in terms of Lagrangian Mechanics.

Noether’s Theorem

Now we come to one of the most profound theorems of mechanics which will make clear the relation between symmetry and conservation laws. Noether’s Theorem states that if you can find a symmetry in the Lagrangian of a system, then that system has a conserved quantity generated by that symmetry.

It’s important to make clear that one can derive Noether’s Theorem purely as a mathematical result. It simply follows from the mathematics of partial derivatives and the way we’ve formulated Lagrangian mechanics.

Now the exciting bit is that when we look at our Lagrangian for a free particle, we can see that there are two symmetries in it. Neither time nor position appear explicitly in the Lagrangian, and so the Lagrangian for a free particle is both time-symmetric and position-symmetric.

So this means that there are two conserved quantities for a free particle, one associated with time-symmetry and the other with position, or rather, space-symmetry. It just so turns out that these conserved quantities are precisely the energy and momentum of the free particle!

For time-symmetric Lagrangians we get energy conservation, and for space-symmetric Lagrangians we get momentum conservation. These conservation laws that may have seemed arbitrary when introduced to you at high-school are actually consequences of symmetry!

As you get deeper into the study of modern physics you find that this is not just a one-off quirk, but that actually symmetry lies at the heart of all our fundamental theories. It might not be too extravagant to claim that fundamental physics is just the study of symmetry in nature.

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