I'm looking at the Casimir effect calculation. As I understand it, it says that because wavelengths can only take on discrete values between two sheets there is a difference in the energy of the vacuum. Because in a vacuum the wavelengths can take on any value. Hence it is the difference between the sum and the integral.

$$E \propto \int\limits_0^\infty{x^3}dx - \sum_{x=1}^\infty x^3 \\ = \lim_{\alpha\rightarrow 0} \left(\int\limits_0^\infty{x^3}e^{-\alpha x} dx - \sum_{x=1}^\infty x^3e^{-\alpha x}\right)\\ = \frac{1}{120}$$

using regularisation.

In Bosonic string theory there is also the sum $\sum n$ where we add up all the modes of the string. But in this case what are we substracting? With analogy to the Casimir calculation it seems like we are comparing it to a slice of an infinite string which can take on any modes which would give an integral. What does this mean?

My interpretation is that in the absence of D-branes the vacuum can have infinite length strings. And that with a pair of D-branes (like the Casimir planes) an open string has a discrete modes of vibration. But this interpretation breaks down when we think of closed strings. What is the interpretation?

I'm looking at the Casimir effect calculation. As I understand it, it says that because wavelengths can only take on discrete values between two sheets there is a difference in the energy of the vacuum. Because in a vacuum the wavelengths can take on any value. Hence it is the difference between the sum and the integral.

$$E \propto \int\limits_0^\infty{x^3}dx - \sum_{x=1}^\infty x^3 \\ = \lim_{\alpha\rightarrow 0^+} \left(\int\limits_0^\infty{x^3}e^{-\alpha x} dx - \sum_{x=1}^\infty x^3e^{-\alpha x}\right)\\ = \frac{1}{120}$$

using regularisation.

In Bosonic string theory there is also the sum $\sum n$ where we add up all the modes of the string. But in this case what are we substracting? With analogy to the Casimir calculation it seems like we are comparing it to a slice of an infinite string which can take on any modes which would give an integral. What does this mean?

My interpretation is that in the absence of D-branes the vacuum can have infinite length strings. And that with a pair of D-branes (like the Casimir planes) an open string has a discrete modes of vibration. But this interpretation breaks down when we think of closed strings. What is the interpretation?

I'm looking at the Casimir effect calculation. As I understand it, it says that because wavelengths can only take on discrete values between two sheets there is a difference in the energy of the vacuum. Because in a vacuum the wavelengths can take on any value. Hence it is the difference between the sum and the integral.

$$E \propto \int\limits_0^\infty{x^3}dx - \sum_{x=1}^\infty x^3 \\ = \lim_{\alpha\rightarrow 0} \left(\int\limits_0^\infty{x^3}e^{-\alpha x} dx - \sum_{x=1}^\infty x^3e^{-\alpha x}\right)\\ = \frac{1}{12}$$

using regularisation.

In Bosonic string theory there is also the sum $\sum n$ where we add up all the modes of the string. But in this case what are we substracting? With analogy to the Casimir calculation it seems like we are comparing it to a slice of an infinite string which can take on any modes which would give an integral. What does this mean?

My interpretation is that in the absence of D-branes the vacuum can have infinite length strings. And that with a pair of D-branes (like the Casimir planes) an open string has a discrete modes of vibration. But this interpretation breaks down when we think of closed strings. What is the interpretation?

I'm looking at the Casimir effect calculation. As I understand it, it says that because wavelengths can only take on discrete values between two sheets there is a difference in the energy of the vacuum. Because in a vacuum the wavelengths can take on any value. Hence it is the difference between the sum and the integral.

$$E \propto \int\limits_0^\infty{x^3}dx - \sum_{x=1}^\infty x^3 \\ = \lim_{\alpha\rightarrow 0} \left(\int\limits_0^\infty{x^3}e^{-\alpha x} dx - \sum_{x=1}^\infty x^3e^{-\alpha x}\right)\\ = \frac{1}{120}$$

using regularisation.

In Bosonic string theory there is also the sum $\sum n$ where we add up all the modes of the string. But in this case what are we substracting? With analogy to the Casimir calculation it seems like we are comparing it to a slice of an infinite string which can take on any modes which would give an integral. What does this mean?

My interpretation is that in the absence of D-branes the vacuum can have infinite length strings. And that with a pair of D-branes (like the Casimir planes) an open string has a discrete modes of vibration. But this interpretation breaks down when we think of closed strings. What is the interpretation?