A jumble of basic SUSY questions (i.e. a bit confused and seeking help)

[shirosato]

So, I've been tasked to learn SUSY in a small time span (few weeks) and I've made some progress but I'm a little stumped on how deeply I should go into things while still making reasonable progress (get to phenomenology, etc).

First of all, most primers seem to gloss over the algebra or go through it quickly, while it seems quite important. I suppose its easier to state my jumble of questions in point form.

1. (perhaps silly question) Is it necessary to know in detail how to derive a the algebra of symmetry generators of a theory? After some work, I learned methods how to do this given the symmetry transformations of a field but it seems quite non-trivial and easy to forget.

2. I understand that the components of the supercharges anti-commute with each other, and this implies that a successive application would annihilate a state. Is there a deeper obvious physical meaning to this?

3. What exactly is the operator $$\sigma^{\mu}P_{\mu}$$? I understand the Pauli matrices generate SU(2) rotations and the 4-momentum generates ST-translation, but physically, what does the projection of them on each other generate?

4. On a related note $$\{Q_{\alpha},Q_{\beta}^{\dagger}\}=(\sigma^{\mu})_{\alpha \beta}P_{\mu}$$ says that some combination of SUSY applications will equal the above mentioned operator, so some translation and rotation. Again, what is the physical implication of this?

5. Does a multiplet consist of states or fields? I understand you can build up states by applying field operators to the vacuum, but I never quite got a good definition of a multiplet despite seeing them and using them quite a bit (I know its states/fields are related by a transformation such as gauge or a SUSY trans.). This caused problems when trying to understand supermultiplets, representations of the superalgebra.6. How the heck to all these particle theorists learn all this stuff to such a deep degree? I've been spending quite a bit of time on this and I'm still scratching my head.

Thanks a bunch,
Hiro

 

[fzero]

1. (perhaps silly question) Is it necessary to know in detail how to derive a the algebra of symmetry generators of a theory? After some work, I learned methods how to do this given the symmetry transformations of a field but it seems quite non-trivial and easy to forget.

It's pretty important, since symmetries are very important for understanding the structure of a theory. Being familiar with the derivations for standard examples can be very helpful when you're learning something new. For example, being very familiar with the Lorentz and Poincare groups is invaluable for learning about SUSY.

2. I understand that the components of the supercharges anti-commute with each other, and this implies that a successive application would annihilate a state. Is there a deeper obvious physical meaning to this?

It's basically the reason that in a minimally supersymmetric theory particles and their superpartners come in pairs. Roughly (the actual algebra is a bit more complicated) if you take a bosonic state, acting with the supercharge gives you the fermionic superpartner. Acting again with the supercharge annihilates the state without giving a new state.

3. What exactly is the operator $$\sigma^{\mu}P_{\mu}$$? I understand the Pauli matrices generate SU(2) rotations and the 4-momentum generates ST-translation, but physically, what does the projection of them on each other generate?

The Lorentz group in 4d is $$SO(3,1)$$. There is a mathematical relation between this group and $$SL(2,\mathbb{C})$$, which is the group of 2x2 complex matrices with unit determinant. The set of matrices $$\sigma^\mu = (1,\sigma^i)$$ provides a map from 4-vectors to 2x2 matrices that realizes this group correspondence. It's a convenient formalism, especially when discussing spinors, which can be naturally expressed as 2-vectors in this representation.

4. On a related note $$\{Q_{\alpha},Q_{\beta}^{\dagger}\}=(\sigma^{\mu})_{\alpha \beta}P_{\mu}$$ says that some combination of SUSY applications will equal the above mentioned operator, so some translation and rotation. Again, what is the physical implication of this?

Several authors refer to the relation above by saying that the supercharges are a kind of square root of the energy operator. In any case, an anticommutator of that form is essential for the supersymmetry algebra to contain the Poincare algebra as a subalgebra. This is essential for supersymmetry to be a physically possible extension of the observed symmetries of nature.

5. Does a multiplet consist of states or fields? I understand you can build up states by applying field operators to the vacuum, but I never quite got a good definition of a multiplet despite seeing them and using them quite a bit (I know its states/fields are related by a transformation such as gauge or a SUSY trans.). This caused problems when trying to understand supermultiplets, representations of the superalgebra.

A multiplet can refer to either states or fields. States are the basic objects, while quantum fields are operators that act on the vacuum state to create particle states. If you're having trouble thinking about gauge or SUSY multiplets, you might want to review the angular momentum algebra in regular QM. The spin multiplets you obtain there, labeled by the $$(j,m_j)$$, illustrate most of the mathematical and physical concepts of the organization of the spectrum of an algebra into multiplets.

6. How the heck to all these particle theorists learn all this stuff to such a deep degree? I've been spending quite a bit of time on this and I'm still scratching my head.

It's mostly about spending time and seeing many examples from basic QM on up, and making the connections between familiar concepts.

 

[shirosato]

Thank you. Those answered clarified things quite a bit and I even agree quite a bit with your answer to question 6. I'm a 2nd year grad student in BSM pheno and am starting to realize 1) the large variance in knowledge over the range of topics in HEP theory among professional theorists 2) the learning curve from a good understanding of undergraduate physics to doing research in HEP theory.

What I realized impeded my understanding of things was the lack of understanding of basic formalism, namely the Hilbert space of QFT and what not.

1. In NRQM, the Hilbert space is straightforward. The S.E. is a linear equation, and its solutions form a Hilbert space and the wave-function is some vector in this space, a superposition of the basis vectors. If we ever have confusion in representation-free formalism, we can easily use the position or momentum basis to easily see what we're doing in terms of functions.

From what I understand, the Hilbert space of QFT is a Fock space. The Fock space is composed of states of definite particle number and are the eigenvectors of the theory's Hamiltonian. I suppose then, the state of the system would be some vector in the Fock space, the equivalent of a wave function? What I don't understand exactly, is if perturbation theory in general is embedded in this idea, since for interacting theories, Fock states are constructed by the free Hamiltonians of particles acting on the vacuum state if I understand correctly. If the interactions are strong, we lose linearity, and anything I say beyond this, will make me sound totally noobie. Since it is a relativistic theory, we lose our position basis and everything is written in momentum. I understand that physical observables are always in the form of amplitudes, the basic quantity of QFT, but is there anything more explicit than |k,k',..>? Perhaps I have some basic misunderstanding of a many-particle system/formalism, or the role of states, measurements, and Hilbert space in QFT?

I do know my understanding of all of this is weak at best, but its hard to find a clear explanation of all these things and I think my supervisor may not want this type of conversation with me.

2. On a related note, I was wondering how the supercharges exactly act on states. The supercharges have a spinor index of $$\alpha=1,2$$. I (might) have read, that one component lowers spin, and the other raises, while the Hermitian conjugate, undoes this action.

For example:

$$Q_{\alpha}Q_{\beta}|boson>=Q_{\alpha}|fermion>_{\beta}=?$$

This object has four components, with two being zero due to anti-commutation. I don't know how exactly to interpret this and would not like to proceed to learning the pheno with such weak basic understanding. Where am I going wrong?

Thank you again,
Shiro

 

[fzero]

What I realized impeded my understanding of things was the lack of understanding of basic formalism, namely the Hilbert space of QFT and what not.

1. In NRQM, the Hilbert space is straightforward. The S.E. is a linear equation, and its solutions form a Hilbert space and the wave-function is some vector in this space, a superposition of the basis vectors. If we ever have confusion in representation-free formalism, we can easily use the position or momentum basis to easily see what we're doing in terms of functions.

From what I understand, the Hilbert space of QFT is a Fock space. The Fock space is composed of states of definite particle number and are the eigenvectors of the theory's Hamiltonian. I suppose then, the state of the system would be some vector in the Fock space, the equivalent of a wave function? What I don't understand exactly, is if perturbation theory in general is embedded in this idea, since for interacting theories, Fock states are constructed by the free Hamiltonians of particles acting on the vacuum state if I understand correctly. If the interactions are strong, we lose linearity, and anything I say beyond this, will make me sound totally noobie. Since it is a relativistic theory, we lose our position basis and everything is written in momentum. I understand that physical observables are always in the form of amplitudes, the basic quantity of QFT, but is there anything more explicit than |k,k',..>? Perhaps I have some basic misunderstanding of a many-particle system/formalism, or the role of states, measurements, and Hilbert space in QFT?

The Hilbert space is the space of states of a single particle. An n particle state lies in a direct product of single particle Hilbert spaces. For a boson, this is $$H \otimes \cdots \otimes H \equiv H^{\otimes n}$$, while for fermions we must take a completely antisymmetric product. The Fock space is the direct sum of all n particle spaces, $$F = \oplus^\infty_{n=1} H^{\otimes n}$$.

In perturbation theory, we assume that the states far from the interaction region are well-described by free particles. That's why for instance we use plane waves to describe the asymptotic fields in a scattering calculation. For a strongly-coupled system like the interior of a nucleus, we can't really do this, which is one reason exact results are hard to obtain.

I do know my understanding of all of this is weak at best, but its hard to find a clear explanation of all these things and I think my supervisor may not want this type of conversation with me.

2. On a related note, I was wondering how the supercharges exactly act on states. The supercharges have a spinor index of $$\alpha=1,2$$. I (might) have read, that one component lowers spin, and the other raises, while the Hermitian conjugate, undoes this action.

For example:

$$Q_{\alpha}Q_{\beta}|boson>=Q_{\alpha}|fermion>_{\beta}=?$$

This object has four components, with two being zero due to anti-commutation. I don't know how exactly to interpret this and would not like to proceed to learning the pheno with such weak basic understanding. Where am I going wrong?

Thank you again,
Shiro

We can't really talk about representations of SUSY without also talking about representations of the Poincare symmetries. Representations of the Poincare algebra are labeled by the mass, $$M$$ and the spin quantum numbers, $$s, m_s$$. So we can write those states as $$| M, s,m_s\rangle$$.

Now because $$Q_\alpha^2 = 0$$, the only non-zero component of $$Q_\alpha Q_\beta$$ is $$Q_1 Q_2$$. Let's take an arbitrary state and act with this,

$$| \Omega_s\rangle = Q_1 Q_2 | M, s,m_s\rangle .$$

$$| \Omega_s\rangle$$ is annihilated by both $$Q_\alpha$$ :

$$Q_1 | \Omega_s\rangle =Q_2 | \Omega_s\rangle =0.$$

In viewing the $$Q_\alpha$$ as ladder operators, $$| \Omega_s\rangle$$ is the lowest-weight state. The rest of the states in the SUSY multiplet can be constructed by acting with the conjugate operators $$Q^\dagger_{\dot{\alpha}}$$ as raising operators:

$$| \Omega_s\rangle$$
$$Q^\dagger_{\dot{1}} | \Omega_s\rangle,~ Q^\dagger_{\dot{2}} | \Omega_s\rangle$$
$$Q^\dagger_{\dot{1}} Q^\dagger_{\dot{2}} | \Omega_s\rangle .$$

The final state is our highest-weight state, as further action with $$Q^\dagger_{\dot{\alpha}}$$ gives zero.

To look at the states in the multiplet a bit more carefully we note that when we act with $$Q^\dagger_{\dot{\alpha}}$$, which has spin 1/2, on a state of spin s, the rules for adding spins tell us the resulting state will have spin $$s\pm 1/2$$. To see how this works, we can look at the scalar multiplet, where

$$| \Omega_s\rangle \sim | m, 0,0\rangle$$

Then

$$Q^\dagger_{\dot{1}} | \Omega_s\rangle,~ Q^\dagger_{\dot{2}} | \Omega_s\rangle \sim | m, \tfrac{1}{2},\pm \tfrac{1}{2} \rangle$$
$$Q^\dagger_{\dot{1}} Q^\dagger_{\dot{2}} | \Omega_s\rangle \sim | m, 0,0\rangle .$$

The two scalar fields comprise a complex scalar, while the two fermionic states comprise a Majorana fermion.