About symmetry operation and a corresponding unitary transformation

[dream_chaser]

For each symmetry operation R acting on a physical system,there is a corresponding unitary transformation U(R).
But what is the principle for such relation?
an example is that : for a continuous symmetry ,we can choose R infinitesimally close to the identity ,R=I+eT ,and the U is close to I,
U=I-ieQ+O(e^2)
another example is that R=(I+Tx/N)^N then U(R)=(I+iQx/N)^N
but i still can not figure out the relation between R and U(R)

 

[dextercioby]

The physical system is modeled (in the absence of relativistic effects) by the $\mathbb{R}^{3}$. In this space symmetry transformations are defined as simply coordinate transformations, see the examples of translations and rotations. So "R" acts in $\mathbb{R}^{3}$, while U(R) acts in an abstract separable Hilbert space which is associated to any quantum system.

 

[strangerep]

For each symmetry operation R acting on a physical system
there is a corresponding unitary transformation U(R).
But what is the principle for such relation?

Not sure what question you're really asking, so I'll just say this...

Quantization basically means representing all of physics in a Hilbert
space. There's a tendency to think that the usual Minkowski spacetime
is objectively "real", and Hilbert space is something else. But they're
both just representation spaces, though it turns out that more
phenomena can be accounted for satisfactorily if we concentrate
on using Hilbert space.

The (dodgy, non-rigorous) passage from a classical theory formulated
on Minkowski spacetime, to a quantum theory formulated on a Hilbert
space is guided by the symmetry group(s) applicable in the classical
theory. That means that if a certain symmetry is represented on
Minkowski space, there must also be a representation of that same
symmetry group on the Hilbert space (else the latter is not a suitable
representation space for a quantized version of that classical theory).

Operators on Hilbert space must be unitary (to preserve the Hilbert
space's inner product between state vectors). Hence the symmetry must
be represented on Hilbert space in terms of unitary operators acting on
its state vectors

an example is that : for a continuous symmetry ,we can choose R
infinitesimally close to the identity ,R=I+eT ,and the U is close to I,
U=I-ieQ+O(e^2)
another example is that R=(I+Tx/N)^N then U(R)=(I+iQx/N)^N
but i still can not figure out the relation between R and U(R)

Both representations must satisfy the same commutation relations
from the abstract group.

Is that what you were asking?

- strangerep.

 

[damgo]

Hi dreamchaser, welcome to PFs!

I'm not sure just what you're asking here... do you want a proof that such a relation exists, a recipe to get U(R), what?

The theorem you're talking about is true in the context of quantum mechanics. In QM we represent the state of a system by a ray in Hilbert space. Let's suppose there is some symmetry R of the physical system. Then if we apply R to the physical system, we get a new physical system, which will be represented by a different ray in Hilbert space. So there has to be some operator U(R) which acts on rays in Hilbert space, where $$U(R) |\psi >$$ is the state of the system after it's been through the transformation R, if the system was originally in the state $$| \psi >$$.

Wigner famously showed that U(R) had to be either a unitary linear operator or an antilinear, antiunitary one. (For most symmetries of interest, it's the former.) The proof isn't trivial; if you want to see it, there's a good version of the proof in volume 1 of Weinberg QFT book -- Chapter 2, appendix A, I believe.

What U(R) is (for a given R) depends on the quantum system you're considering. There isn't a simple straightforward 'formula' or such to get U(R) from R. But there's a great deal of machinery that has been developed which helps us to classify the possible U(R)'s. This is sometimes called representation theory; since most groups R of interest are continuous (Lie) groups it's often covered along with Lie groups & algebras. For a given symmetry group R, there are a variety of different possible U(R)'s for different objects; we call these representations of R. Different objects can "transform as" different representations of R.

For example, for the 3d rotation group SO(3), we can label the possible representations by a nonnegative half-integer or integer j, ie j=0,1/2,1,3/2, ... These correspond to objects of different total angular momentum j. For an object in the j=0 representation, U(R) is just the identity. For an object in the 'spinor' representation j=1/2, we have $$U(R) = \exp(-i \theta_j \sigma_j)$$ with theta_j the parameters of the rotation and sigma_j the Pauli matrices.

 

[smallphi]

For each symmetry operation R acting on a physical system,there is a corresponding unitary transformation U(R).
But what is the principle for such relation?
an example is that : for a continuous symmetry ,we can choose R infinitesimally close to the identity ,R=I+eT ,and the U is close to I,
U=I-ieQ+O(e^2)
another example is that R=(I+Tx/N)^N then U(R)=(I+iQx/N)^N
but i still can not figure out the relation between R and U(R)

Excellent question.

Let's consider for definiteness rotations. Imagine you have a coordinate system with axes defined by your measuring aparatus, you could measure momentum in certain direction or spin projection in certain direction etc. Initially your quantum system is at angle $$\theta = 0$$ with respect to the coordinate system and its quantum state is the vector $$|\theta = 0>$$ in some Hilbert space built up from all possible states of the system. This state can be any state in which we prepare the system.

Now let's rotate your quantum system by angle $$\theta_1$$ with respect to the aparatus. Although nothing internallly changed in your system, it's relation (angle) to the measuring apparatus changed and so is its state with respect to the measuring aparatus/coordinate system. We can describe that state with a vector $$|\theta_1>$$ in the same Hilbert space since its the same type of system - we don't need different Hilbert space for every rotated version of the system.

Now imagine we repeat the experiment by rotating the system from angle zero to angle $$\theta_2$$ and state $$|\theta_2>$$.

A third experiment is rotating the system from angle zero to angle $$\theta_1+\theta_2$$ and state $$|\theta_1+\theta_2>$$.

So in general when you rotate the system with respect to the measuring aparatus, the measured system quantum state changes although we did nothing internally on the system to change it, only the relation to the measuring apparatus changed and the state is defined with respect to the aparatus. We can imagine that change in the quantum state is effected by some operator in the Hilbert space, cause that is what operators do, they take a state and change it to another state. Call this operator U and it will depend on the angle of rotation:

$$|\theta_1> = U(\theta_1)|\theta=0>$$
$$|\theta_2> = U(\theta_1)|\theta=0>$$
$$|\theta_1+\theta_2> = U(\theta_1+\theta_2)|\theta=0>$$

Now comes the physical assumption that relates those rotated quantum states. Our physical intuition tells us that it shouldn't matter if we rotate the system directly to angle $$\theta_1+\theta_2$$ or we first rotate by angle $$\theta_1$$ and then rotate by extra angle of $$\theta_2$$. In the language of the quantum states that mean if we take the state $$|\theta_1>$$ and rotate it by an extra angle of $$\theta_2$$, the result should be up to a phase factor the state $$|\theta_1+\theta_2>$$:

$$U(\theta_2)|\theta_1> = exp(i\phi) |\theta_1 + \theta_2>$$.

In terms of the non-rotated state:

$$U(\theta_2)U(\theta_1)|\theta = 0> = exp(i\phi) U(\theta_1+\theta_2)|\theta = 0>$$

Since the above is true for any initial state $$|\theta = 0>$$ in the Hilbert space, we get the operator equality:

$$U(\theta_2)U(\theta_1) = exp(i\phi) U(\theta_1+\theta_2)$$

A very similar equality is satisfied by the matrices R that we use to rotate coordinates:

$$R(\theta_2)R(\theta_1) = R(\theta_1+\theta_2)$$

Because of the similarity of the the last two equations we say that the operators U in the Hilbert space represent the rotations R in coordinate space. The exact mathematical term is, the operators U form a ray or projective representation of the rotations R. Again that derivation was based on our physical assumption that rotation by sum of angles must produce the same physical result as rotation by the first angle and additional rotation by the second angle.

Now comes the Wigner theorem that all of the U operators must be unitary or anti-unitary if we want the U operator to preserve the probabilities i.e. all possible matrix elements squared:

$$|(U|A>, U|B>) |^2= |(|A>, |B>)|^2$$

for any two states |A> and |B> in the Hilbert space. We want that because a rotation of the system should not change the relative probabilities between different states.


Mathematicians have enlisted the possible representations U of the rotations R in finite or infinite dimensional Hilbert spaces called correspondingly finite or infinite representations. Which representation a given system would choose is up to mother nature, it can't be decided in advance on theoretical grounds. Only experiment can probe the choice realized for a given system.