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Unitary Groups is the most mysterious thing for me when studying physics. All my physics endeavor ends when author starts talking about unitary groups. This is often the case because in a lot of the texts or papers the author would throw terms like $SU(3)$, $SU(5)$ without any background motivation.

When you search for some more answers online, you will often get a purely mathematical treatment (i.e. Wiki article: http://en.wikipedia.org/wiki/Special_unitary_group) or something related to quantum field theory. Now remind me how many years of studying physics will one reach QFT?

Could someone qualitatively explain or provide a good article that motivates the concept of unitary group with examples drawn from ... every day experience or basic physics. Could someone at least explain what an element of an unitary group physically mean?

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3 Answers 3

To develop physical intuition behind unitary groups, it's helpful to have intuition for the notion of unitary transformations in general since unitary groups are just groups consisting of unitary transformations.

It helps to start with mathematical objects that are a bit more "familiar" in classical physics -- rotations, which are also known as orthogonal transformations. Why? Because unitary transformations are basically the same thing as orthogonal transformations when you start using complex numbers (complex vector spaces) instead of real numbers.

Recall that rotations in three spatial dimensions are simply transformations that don't change the dot products between vectors; \begin{align} R\mathbf v\cdot R\mathbf w = \mathbf v \cdot \mathbf w. \end{align} Recall also that in classical physics, rotations are special because observers with axes that are rotated relative to each other will measure the same physics. In fact, the rotation group is a subgroup of the Lorentz group which connects the measurements of two inertial observers. The same sort of idea, that a certain kind of transformation leaves something about what you would measure in the real world, can be used to understand unitary transformations as well.

Suppose that you are working with a vector space over the complex numbers where there is a notion of "dot product." In this more general context, such a product is usually called an inner product, and in this context we can again consider transformations that don't change the inner products between vectors \begin{align} \langle Uv, Uw\rangle = \langle v,w\rangle \end{align} and these guys are called unitary transformations.

But now you could ask "well who the heck cares about this stuff in physics. I mean, what does this have anything to do with anything in the physical world?" In my view, the best answer for this comes from quantum mechanics. Quantum mechanics has at its very foundations complex vector spaces with inner products since the (pure) states of quantum systems are vectors in complex vector spaces, and the inner products of these vectors allow one to compute probabilities for certain measurement outcomes.

In this context, unitary transformations become very physical and very important; they can be thought of as symmetry transformations on the quantum system because they preserve the inner product which determines measurement probabilities. In other words, unitary transformations don't change measurement outcomes in quantum mechanics. This is kind of like how rotations don't change what an observer measures in classical physics.

For more details and a deeper understanding of what I'm talking about here with respect to unitary transformations and their significance in describing symmetry in quantum mechanics, consider reading about Wigner's Theorem on symmetries in quantum mechanics.

You may also find the following to be helpful:

What is a symmetry of a physical system?

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Thanks, this is a good starting point for me. where do you guys learn these stuff? All the physics book I've ever owned covers generally everything from newtonian physics to thermodynamics to quantum physics but without going into so much detail. Occasionally I'd read up a book on a specific subject i.e. Jackson - ED, Griffith - QM, but nobody goes into this type of stuff. Any good recommendations? –  Illegal Immigrant Aug 31 at 0:43
    
@IllegalImmigrant Good graduate texts on quantum mechanics often talk about this stuff. Not that many physics books that I know of talk about Wigner's Theorem unfortunately. –  joshphysics Aug 31 at 0:45
    
Wigner has a collection of essays that you might find helpful. The book is called "Symmetries and Reflections". It's deep, but not too math-y. The whole first section is on symmetries in physical systems and how it leads us to discovering laws of nature in the first place. amazon.com/Symmetries-Reflections-Scientific-Eugene-Wigner/dp/… –  jjgoings Aug 31 at 2:18
    
A follow up for your answer: Why is it useful to consider the symmetry transformations for a system? –  BMS Aug 31 at 9:44
    
@BMS You should write an answer that addresses that follow up. You have my +1 if you do. –  joshphysics Sep 2 at 2:29

As you probably know, in quantum mechanics the normalization of state vectors (resp. wave functions) represents the normalization or probabilities. Written formally, the expectation value of observable $A$ in state $\left|\psi\right\rangle$ is $\left\langle\psi\right|A\left|\psi\right\rangle$, and normalization basically ensures that the expectation value of a constant is that constant, $\left\langle\psi\right|1\left|\psi\right\rangle = \left\langle\psi\middle|\psi\right\rangle=1$. This ultimately is nothing but the demand that probabilities have to add up to $1$.

Now when doing transformations, for example time evolution, you certainly want to ensure that this normalization condition is preserved. Moreover you are generally interested in invertible operations (for example, time evolution of closed quantum systems is invertible). Note that this extra assumption is only needed for infinite-dimensional Hilbert spaces; for finite-dimensional Hilbert spaces, already the demand of norm conservation is sufficient.

Now if we call (in anticipation of the result) the transformation $U$, then the transformed (in the case of time evolution, later) state reads $\left|\phi\right\rangle = U \left|\psi\right\rangle$. Now the normalization condition reads $$\left\langle\phi\middle|\phi\right\rangle = \left\langle\psi\right|U^\dagger U\left|\psi\right\rangle$$ and since that one has to hold for all $\left|\psi\right\rangle$, it follows that $U^\dagger U = 1$, which, since $U$ was supposed to be invertible, means $U^\dagger = U^{-1}$. But that's exactly the definition of an unitary transformation.

OK, so now we have unitary transformation, but what is an unitary group? Well, an unitary group is simply a set of unitary transformations with certain properties:

  • It contains the identity transformation (that is, doing nothing; this certainly doesn't change the normalization, and therefore is an unitary transformation).
  • For ever two transformations, it also contains their product (applying one operation after the other; it's not hard to check that the product of two unitary transformations is again an unitary transformation, as not changing the norm twice does still not change it).
  • For every transformation, it also contains the inverse transformation (the inverse transformation of an unitary transformation is quite obviously also a unitary transformation).

Let's look at an example how finite unitary groups enter quantum mechanics long before you get to quantum field theory.

The example is spin: You might already know that the electron has a quantum property called "spin" which is sort of an angular momentum that is not connected to rotation, but an intrinsic property of the electron. When measured in any direction, you can get only one of two values, "spin up" (usually denoted by $\left|\uparrow\right\rangle$) and "spin down" (usually denoted by $\left|\downarrow\right\rangle$). Therefore, according to the superposition principle (and for the moment ignoring the space dependence of the wave function), the most general spin state of an electron is $$\left|\psi\right\rangle = \alpha\left|\uparrow\right\rangle + \beta\left|\downarrow\right\rangle,\quad \left|\alpha\right|^2+\left|\beta\right|^2=1$$ where the condition is due to the normalization constraint.

However, remember that the "spin up" and "spin down" states are defined relative to a certain direction. But we know that nature is isotropic, that is, it doesn't have a preferred direction (a specific experiment may, of course, impose a special direction, for example through an external magnetic field). Therefore it must be possible to describe the very same state with the "spin up" and "spin down" states of any other direction (which is the same as to rotate the frame of reference). Say, the "spin up" and "spin down" states corresponding to that other direction are $\left|\uparrow'\right\rangle$ and $\left|\downarrow'\right\rangle$. Then we have $$\left|\psi\right\rangle = \alpha \left|\uparrow\right\rangle + \beta\left|\downarrow\right\rangle = \gamma \left|\uparrow'\right\rangle + \delta \left|\downarrow'\right\rangle$$ Quite obviously, to support superpositions, the transformation from $(\alpha, \beta)$ to $(\gamma, \delta)$ has to be linear; also it clearly has to be invertible, because we can read that equation in both directions. Finally, we of course have to obey the normalization conditions in both descriptions.

So to get from $(\alpha, \beta)$ to $(\gamma, \delta)$ we have to apply an invertible, norm-reserving linear operation, that is, an unitary operation. The set of unitary operations on dimension two (we have two numbers here!) is known under the name $U(2)$. However it turns out we do not need all of those. The reason is that a global phase on the state vector does not change the physical state. Now unitary transformations always have a determinant with absolute value $1$. Since the phase does not matter, we can always choose it so that the determinant is actually $1$. Indeed, those transformations, known as "special unitary operations", again form a group, $SU(2)$. And it turns out that we need all of $SU(2)$ to describe rotations of the spin.

At this point, you'll probably also not be surprised to hear that there's a close relation of the group $SU(2)$ with the group $SO(3)$ of rotations in space. Indeed, for every $SU(2)$ transformation, there exists exactly one corresponding $SO(3)$ transformation. The reverse is not true; rather for each $SO(3)$ transformation there are two $SU(2)$ transformations. However those two transformations only differ in sign, and therefore don't make a physical difference (however it has very deep implications that you cannot get rid of this minus sign, unlike you can with the rest of the phase when going from $U(2)$ to $SU(2)$).

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If you're comfortable with normed complex vector spaces (like the good 'ol Hilbert spaces of quantum mechanics), then unitary groups are extremely natural. They are the orthogonal groups of the complex world: length-preserving maps. Specifically, $U(n)$ is the set of all functions on $\mathbb{C}^n$ that preserve the norm $u^\dagger u$ for all $u\in\mathbb{C}^n$.

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