Consider a real scalar $\phi(x,t)$ with mass $m$ in $1+1$ dimensional spacetime, described by the 2d free Klein-Gordon action. $\phi(x,t)$ lives on an interval $0 \leq x \leq L$, and is subject to the Dirichlet boundary conditions: $$\phi(0,t) = \phi(L,t) = 0.$$ Quantize this system and show that the formal (divergent) expression for the vacuum energy is $$E_0 = \sum_{n = 1}^\infty \frac{E_n}{2} = \sum_{n = > 1}^\infty \frac 12 \sqrt{(\frac{\pi n}{L})^2 + m^2}.$$

I know how to quantize free Klein Gordon equation. However, in the above, there is the boundary condition. Is it possible to just quantize the free Klein Gordon equation and apply the boundary condition? I am very confused...

  • $\begingroup$ In this case imposing the constraints and second quantization commute with each other, but probably they don't in general... $\endgroup$ – Ryan Thorngren May 2 '18 at 18:34

Hint: when you are quantizing KG, you use Fourier transform integrals. Here with the Dirichlet boundary conditions, only a certain discrete set of modes is allowed. Your job is to repeat the usual 2nd quantization but use Fourier series instead of Fourier integrals. Can you obtain the Fock space? What is the ground state energy of your oscillators?

Bonus question: does it make sense to define a momentum operator? What about the boost generator?

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