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I would like to check if my understanding of the definition of a particle is right. I also have some questions on things I still don't understand.

On pages 109-111 of Schwartz, Quantum field theory and the standard model he defines particles, here is what I understood from it.


We consider a state of our Hilbert space :$|\psi\rangle$. We consider the group of Poincaré transformations. A particle can have different characteristics : its momentum $p$, its mass $m$, its spin, etc.

Two states $|\psi\rangle$ and $|\psi'\rangle$ represent the same particle if we can go from one to the other one by doing a Poincaré transformation.

If it is not possible, then those states represent two different particles.

Indeed, for example $|p_1,m\rangle$ and $|p_2,m\rangle$ represent of course the same particle but that goes at two different speed.

If I am not possible to do a Poincaré transformation linking my two states it means that an "intrinsic" characteristic of my states are different, the mass for example. Thus I deal with two differents particles.

Here is the physical motivation, and now we try to find all the particles using maths.

Mathematically, the tool associated to it is the theory of representations of the Poincaré group. What we have to do is to find the subspaces of the Hilbert space in which $\forall g, ~ D(g)$ is stable, where $D$ is the representation of the Poincaré group. If we do it we will be able to find all the different states that are linked by a Poincaré transformation, and thus all the different particles !

Said differently, we need to find the irreducible representations of the Poincaré group. Each irreducible representation will be associated to a given particle.

Actually we need to find the unitary irreducible representations of the Poincaré group. Unitary because as we deal with Q.M we need to have conservation of the norm of our state.

What Schwartz says is that there is no finite dimension unitary irreducible representation of the Poincaré group. They are all infinite dimensional. This is the motivation for describing the particles as fields.

Wigner showed that there are two indices to index those representations : a real positive number $m$ and a half integer $J$. This will be our definition of mass and spin for particles.

But after Schwartz says that if $J>0$, there are $2*J+1$ independant states in the representation. I see that physically it would correspond to the value of the spin. But I don't understand to what "independant states" we refer. Indeed, from what I understood, I can write my representation like this :

$$ D(g)=\begin{bmatrix}[D_{m_0,J_0}(g)] & ~ & ~ & ~ \\ ~ & [D_{m_1,J_1}(g)] & ~ & ~ \\ ~ & ~ & \ddots & ~ \\ ~ & ~ & ~ & [D_{m_k,J_k}(g)] \\ ~ & ~ & ~ & ~ & \ddots \\ \end{bmatrix} $$

Each diagonal block in this matrix is an irreducible representation (so represent a given particle) that is infinite dimensional. So what are this $2J+1$ independant states ? We have infinite dimensional irreducible representation so it can't be the dimension of the diagonal blocks. I am confused here...

So in summary, my two questions are:

  1. Did I understood well the definition of particle?

  2. What exactly are those $2J+1$ independant states? Aren't we working in an infinite dimensional space?

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The Poincare group has two pieces: translations and Lorentz transformations. The translations commute with each other, so we can simultaneously diagonalize them, leading to wavefunctions proportional to $e^{-ipx}$. If that's the whole story, then we're done: the Lorentz transformations just act by transforming $p$, and we get one state for each particle momentum. This is an infinite-dimensional irreducible representation corresponding to a particle with spin $J = 0$.

More generally, it could be the case that there are multiple states for each value of $p$, e.g. for a particle with spin, so the state is a position-space wavefunction times a spin wavefunction. Wigner's classification essentially says that for a massive particle this is the only thing that can happen, with $J$ indexing the spin. The entire irrep is still infinite-dimensional since we can still have any $p$.

To do a general Poincare transformation, you transform the plane wave part just like you would normally, i.e. rotating it for a rotation and boosting it for a boost. Then you account for how the spin state changes. Schwartz's notation is suppressing the first part.

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  • $\begingroup$ Thank you for your answer. My comment is probably very basic but I want to be sure I understand well what you said. The dimension of the Lorentz and translation groups are both infinite. The dimension of the translation representations is infinite. But the dimension of the lorentz representation is $2J+1$ which is finite. So the dimension of the representations are $+ \infty + 2J+1=+ \infty$, this is what you wanted to say. In the end schwartz says that the representation are infinite dimension because of the translations. If we forget them we have $2J+1$ finite dimension left. $\endgroup$ – StarBucK Feb 4 '18 at 14:47
  • $\begingroup$ So in a global aspect what I said in my post is mainly true ? I didn't do any big mistake in my understanding ? (To be sure) $\endgroup$ – StarBucK Feb 4 '18 at 14:48
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    $\begingroup$ The groups are Lie groups of finite dimension (6 and 4) but infinitely many elements. You can kind of think of the Poincare rep as a finite dimensional Lorentz rep “times” an infinite dimensional translation rep but this is not quite true, since the two factors affect each other. (For example, a rotation of the momentum also rotated the spin.) But I guess you can kind of count the dimension that way. $\endgroup$ – knzhou Feb 4 '18 at 14:51
  • $\begingroup$ Yes indeed I was confused by the number of element of the group which is different from the dimension of the group. Thank you a lot for your answers $\endgroup$ – StarBucK Feb 4 '18 at 15:14

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