What do we mean when we say something transforms "under"....

[AndrewGRQTF]

What do we mean when we are talking about something that transforms under a representation of a group? Take for example a spinor. What is meant by: this two component spinor transforms under the left handed representation of the Lorentz group?

When we talk about something that transforms, shouldn't we first say what object is being transformed and what transformation we are applying to that object, and after that say how/to what it transforms? In the previous example of a spinor, could someone answer the following questions:

What is being transformed:
What is the transformation applied:
How does it transform/To what does it transform:

Also, isn't a representation of a group just an assignment of a matrix to each group element such that the matrices behave under matrix multiplication the same way the abstract group elements behave under the group operation? What is so special about the statement "this transforms under that representation" that makes it descriptive? Can't we act with the representation on anything we want (as long as the dimensions match up)?

If this entire question is based on some misunderstanding that you spotted, then feel free to ignore my question and tell me what I should do to change my wrong state of mind.

 

[king vitamin]

There are probably too many questions here to answer them one-by-one, and additionally I think the different questions have some similarities. In particular, I think this paragraph gets to the heart of all of your questions:

Also, isn't a representation of a group just an assignment of a matrix to each group element such that the matrices behave under matrix multiplication the same way the abstract group elements behave under the group operation? What is so special about the statement "this transforms under that representation" that makes it descriptive? Can't we act with the representation on anything we want (as long as the dimensions match up)?

In some sense you're correct - if I create an $N$-dimensional column vector, I can always formally choose to act on it by left-multiplication of an $N$-dimensional representation of some group. But then one needs to ask whether it makes sense for the individual components in that vector to actually transform that way. If I were to place the temperature, pressure, and volume of some thermodynamic system into a three-vector, and then start applying the usual 3d rotation matrices, as though these quantities rotated into each other if I rotated my system, you would think I was crazy. Those quantities don't change under a spatial rotation! Our physical understanding of them clearly implies that they are scalars under rotation (they transform under the trivial/one-dimensional representation of rotations).

So when we construct multi-dimensional objects and specify how they transform, we're not just matching up dimensionalities. We also need some physical input about how the physical symmetries affect the objects. We know that spinors go to minus themselves under a $2\pi$ rotation, and we also have an understanding of how parity is violated in certain theories we want to describe using spinors. By studying our symmetries mathematically, we end up with objects like the the two-dimensional Weyl spinors which transform in the correct way that we see in experiment. They transform this way by construction, because we want to construct objects which behave like our experiments.

 

[AndrewGRQTF]

There are probably too many questions here to answer them one-by-one, and additionally I think the different questions have some similarities. In particular, I think this paragraph gets to the heart of all of your questions:
In some sense you're correct - if I create an $N$-dimensional column vector, I can always formally choose to act on it by left-multiplication of an $N$-dimensional representation of some group. But then one needs to ask whether it makes sense for the individual components in that vector to actually transform that way. If I were to place the temperature, pressure, and volume of some thermodynamic system into a three-vector, and then start applying the usual 3d rotation matrices, as though these quantities rotated into each other if I rotated my system, you would think I was crazy. Those quantities don't change under a spatial rotation! Our physical understanding of them clearly implies that they are scalars under rotation (they transform under the trivial/one-dimensional representation of rotations).

So when we construct multi-dimensional objects and specify how they transform, we're not just matching up dimensionalities. We also need some physical input about how the physical symmetries affect the objects. We know that spinors go to minus themselves under a $2\pi$ rotation, and we also have an understanding of how parity is violated in certain theories we want to describe using spinors. By studying our symmetries mathematically, we end up with objects like the the two-dimensional Weyl spinors which transform in the correct way that we see in experiment. They transform this way by construction, because we want to construct objects which behave like our experiments.

Thank you very much for your reply. Your example of the vector with components of temperature/pressure/volume is very simple yet beautiful and cleared up a lot of my confusion. I also found an explanation of this in a book: we say an object transforms under a certain representation, if when we apply a transformation to our frame, we observe that the object transforms to another object according to the transformation rules of that representation. What I would like to know is how the transformation to our frame relates to the transformation of the object. Also, what kind of mathematical object is our frame?

 

[Spinnor]

You might find this helpful, I did.

https://arxiv.org/abs/1312.3824

From that,

"... Spinors can be used without reference to relativity, but they arise naturally in discussions of the Lorentz group. One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed. ..."

 

[AndrewGRQTF]

You might find this helpful, I did.

https://arxiv.org/abs/1312.3824

From that,

"... Spinors can be used without reference to relativity, but they arise naturally in discussions of the Lorentz group. One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed. ..."

I have found the answer to my question. In case any other students have the same question, I will write it here.

Take the case of a Dirac spinor $\psi ^\alpha (x)$ where its components are labeled by $\alpha = 1,2,3,4$. After performing a Lorentz transformation $\Lambda$ to our frame, the Dirac spinor transforms like $$\psi ^\alpha (x) \to S[\Lambda]^\alpha\; _\beta \psi ^\beta (\Lambda ^ {-1}x)$$ where $$\Lambda = exp(\frac {1}{2} \Omega _{ab} M^{ab})$$
and $$S[\Lambda] = exp(\frac {1}{2} \Omega _{ab} S^{ab})$$ The $\Omega _{ab}$ are (antisymmetric) coefficients with a,b=0,1,2,3 corresponding to six independent parameters: three of them are the spatial rotation angles and the other three are the boost angles that mix a space coordinate and time. The $M^{ab}$ are the generators of the vector representation of the Lorentz group and the $S^{ab}$ are the generators of the certain representation that the spinor transforms under (a and b label the matrices and not rows or columns). The $S[\Lambda]$ is what acts on the spinor; the transformed spinor shown above is the one we would "see" after we perform a vectorial Lorentz transformation $\Lambda$ to our frame. The way that we find the correct $S[\Lambda]$ associated to a certain $\Lambda$ is to use the same coefficients $\Omega _{ab}$ that we used for $\Lambda$.

The change of the spinor argument from $x$ to $\Lambda ^{-1} x$ is purely due to our frame transformation. It does not mean that the object we are observing truly changed (this is the way scalar fields transform - we call that trivial). The $S[\Lambda]$, however, really does mix up the components of the spinor in a non-trivial way. Whenever we say "an object transforms under a certain representation", we refer to the $S[\Lambda]$ in this problem, or whatever the analogous term in your problem is. Things do not transform for no reason; they transform when we change something, and the way they transform always (obviously) depends on the change we did.