String Theory, Number Operator , Mass of States

[binbagsss]

Homework Statement

I have the following definition of the space-time coordinates

Homework Equations

Working in a certain gauge we can also do:

From which we can find:

Where ##N_{lc} ## sums over the transverse oscillation modes only.

The Attempt at a Solution

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MY QUESTION:

I don't understand the RHS of the first excited state, how does ##N## take integer values? what exactly are the ##\alpha##?

So (here the summation is over all modes, whereas above it is over the transvere oscillation modes only but ignoring this)

so for the first excited state do we have all alpha modes are zero except ##\alpha_{-1}## ?

So Then ## N= \alpha^{j}_{-1} ## ? how is this ##N=1##?

So for the next excited state I expect:

## m^2 \alpha_{-1}^i \alpha_{-1}^j |p^i>=m^2\alpha^{j}_{-2}|p^i>=(2-a) \alpha_{-1}^i \alpha_{-1}^j |p^i>=(2-a) \alpha_{-2}^j |p^i> ## right?

And I am told in my notes that this can be achieved by either ##\alpha_{-1}^i \alpha_{-1}^j |p^i> ## or ##\alpha_{-2}^i##

How does this get ##N=2##? since, well looking at ## \alpha_{-1}^i \alpha_{-1}^j ##, the sum in ##N## is over ##n## not ##i## or ##j## so am I looking at two different number operators here?

For the other expression So Then ## N= \alpha^{j}_{-2} ## ? how is this ##N=2##?

I'm just really confused as you can tel..

Many thanks for your help in advance.

 

[mark726]

Thank you for your question. Let me try to clarify the notation and equations in the forum post.

First, the space-time coordinates are defined as the set of variables x^0, x^1, x^2, and x^3. These variables represent the time and three spatial dimensions.

Next, the equations that follow are related to a specific gauge, which is a choice of mathematical representation in a particular physical theory. In this gauge, the equations involve the summation over transverse oscillation modes only, denoted by N_lc. The notation N_lc is used to represent the number operator, which counts the number of particles in a given state. In this case, the sum over N_lc is taken over all possible transverse oscillation modes.

Now, let's look at the first excited state. The notation α represents the creation and annihilation operators for the transverse oscillation modes. These operators are used to create or destroy particles in a given state. In the first excited state, there is only one particle present, which is represented by the subscript -1. Therefore, the only non-zero term in the sum over N_lc is when N_lc = 1. This is why N = 1 in this case.

Moving on to the next excited state, we see that there are two possible ways to create this state. One way is by using the creation operators α^-1 and α^-1, which corresponds to the expression α^-1 α^-1|p^i>. The other way is by using the creation operator α^-2, which corresponds to the expression α^-2|p^i>. Both of these expressions have N_lc = 2, which is why N = 2 for this excited state.

I hope this helps to clarify the notation and equations in the forum post. If you have any further questions, please don't hesitate to ask.