Notes on string theory: Quantisation – the point particle case


We have already established our notation in the light-cone (LC), and have worked through several prefacing topics prior to quantisation. We can now consider how we might establish our notation in QM.

First, the conjugate variable for time is the Hamiltonian. On the other hand, the conjugate variable for X is the momentum. We can also note,

    \[ \big\{t, X\big\} \rightarrow \big\{H, P\big\} \]

Now that we have defined our conjugate variables, we can write our operators in a fairly intuitive way. In short, to quantise we “impose canonical commutators on the dynamical fields” (p. 17).

    \[ P_{\mu} = -i \frac{\partial}{\partial X^{\mu}} \rightarrow_{commutation \ rel.} [P_{\mu}, X^{0}] = -i \delta_{\mu}^{\nu} \]

From this notice, P_0 = -i \frac{\partial}{\partial \tau} because X^{0}=\epsilon. Furthermore, as Polchinski notes (p.17), i \frac{\partial \psi}{\partial \tau} = H \psi and so it follows H = i \frac{\partial}{\partial \tau}. In the gauge-fixed theory it is found that H = - P_{0}.

From pages 16-17 in Polchinski, what we learn, ultimately, is that to construct the Hamiltonian we must first identify what is the time in the theory. From this we can construct our conjugate variable, relate the conjugate momentum to time, which then allows us to obtain P_0. It is therefore the case that in our theory -P_0 is the Hamiltonian of the system.

Point particle in LC
Before quantising the string, we first study the case of the point particle.

Recall, firstly, that for the point particle there is a worldline time. We denoted this as \tau. There is also \tau on the WS of the string. One advantage, it can be seen, is that we are free to choose whatever tau we want. In the below working, the choice for \tau is to be along X^{+}. So now it is defined as LC time. Expanding the line of logic being built, this means there will be a conjugate momentum to X^{+}. In other words, as given in Polchinski, the parameterisation is fixed in such a way,

    \[ X^{+} = \tau \rightarrow P^{+} \]

And, from this, we can now declare what is the Hamiltonian. It is given below,

    \[ H = -P_{+} = P^{-} \]

So it can be observed, quite nicely in fact, that P^{-} plays the role of the LC energy.

With these matters discussed, and, as noted above, with the parameterisation of the worldline fixed, we can return to the S_{pp}^{\prime} action (p.17). Recall, in Polchinski,

    \[ S_{pp}^{\prime} = \int d\tau [\eta^{-1} \dot{X}^{\mu} \dot{X}_{\mu} - \eta m^2 ] \]

Now, in the light-cone,

    \[ = \frac{1}{2} \int d\tau [-2 \eta^{1} \dot{X}^{+}\dot{X}^{-} + \eta^{-1}\dot{X}^{i}\dot{X}^{i} - \eta m^2] \]

What is interesting, as a consequence of what we’ve developed, is that up to now we have described \tau as an arbitrary parameterisation along the worldline. That remains true. But we have also since agreed to define it along the LC time. And, so,

    \[ X^{+} = \tau \ \ \ \dot{X}^{+} = 1 \]

    \[ \implies \frac{1}{2} \int d\tau [-2 \eta^{-1}\dot{X}^{-} + \eta^{-}\dot{X}^{i}\dot{X}^{i} - \eta m^2] \]

Naively, we may consider \eta, X^{i}:

    \[ P_{\mu} = \frac{\partial \mathcal{L}}{\partial \dot{X}^{\mu}} \]

And, so, we may write for P_{-} and P_{i},

    \[ P_{-} = -\eta^{-1} \implies (P^{+} = \eta^{-1}) \]

    \[ P_{i} = \eta^{-1} \dot{X}^{i} \implies \dot{X}^{i} = \eta \cdot p_{i} = \frac{P_{i}}{P^{+}} \]

Where \eta = \frac{1}{P^{+}}. What is interesting, then, is that we can determine,

    \[ \big\{X^{-}, X^{i} \big\} \implies \big\{P_{-}, P_{i} \big \} \]

Now, we can write the Hamiltonian (where, on p.17 in Polchinski, it is noted that X^{+} is not a dynamical variable),

    \[ H = P_{-}\dot{X}^{-} + P_{i}\dot{X}^{i} - L \]

Where, L = -2\eta^{-1} \dot{X}^{+} \dot{X}^{-} + \eta^{-1} \dot{X}^{i}\dot{X}^{i} - \eta m^2.

And so, we can finally arrive at (1.3.6) in Polchinski.

    \[ H = P_{-}\dot{X}^{-} + P_{i}\dot{X}^{i} - L \]

    \[ = P_{-}\dot{X}^{-} + \frac{P_i P_i}{P^{+}} + \eta^{-1}\dot{X}^{-} - \frac{1}{2}P^{+} \frac{P_i P_i}{P^{+ 2}} + \frac{1}{2} \eta m^2 \]

    \[ = P_{-}\dot{X}^{-} + \frac{P_i P_i}{P^{+}} - P_{-}\dot{X}^{-} - \frac{1}{2}P^{+} \frac{P_i P_i}{P^{+ 2}} + \frac{1}{2} \eta m^2 \]

    \[ \frac{1}{2} \frac{P_i P_i}{P^{+}} + \frac{1}{2P^{+}} m^2 \]

    \[\implies \frac{P_i P_i + m^2}{2P^+} \]

We therefore arrive in line with Polchinski, where H = \frac{P^i P^i + m^2}{2P^+} such that we may also denote H = P^{-}. So,

    \[ \implies 2P^{+}P^{-} - P^{i}P^{i} = m^2 \]

    \[ \implies -2 P^{+}P^{-} + P^{i}P^{i} = -m^2 \]

    \[ \therefore P_{\mu}P^{\mu} = - m^2 \]

To conclude, it is determinable that from classical canonical analysis we can do canonical quantisation by replacing,

    \[ [P_{i}, X^{i}] = -i \delta_{j}^{i} \]

    \[ [P_{-}, X^{-}] = -i \]

When we go ahead and quantise, the states (momentum eigenstates) of the particle “form the complete set” (p.17): | K_{-}, K^{i} \rangle.

In the next post we’ll turn our attention to quantising the string for the first time.

References

Joseph Polchinski. (2005). “String Theory: An Introduction to the Bosonic String“, Vol. 1.