Questions and answers about string theory here.
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Moved from previous thread.
(SR) String Theory Q and A
- Thread starter BenTheMan
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Ben,
I thought the S matrix was by definition unitary regardless of string theory. (Disclaimer: Unfortunately, I am no mathematician)
Possibly related general question:
Does string theory need to be background independent? Has string FIELD theory been completely dropped?
I thought the S matrix was by definition unitary regardless of string theory. (Disclaimer: Unfortunately, I am no mathematician)
Possibly related general question:
Does string theory need to be background independent? Has string FIELD theory been completely dropped?
initial state and final state of interaction of particles.
Well, perhaps I should say that the evolution is goverened by a unitary S matrix.
Well, there is a bosonic string field theory, which is fully background dependant, and has been solved recently by Martin Schnabl (I think I spelled his name right). So the objections that string theory is not background independant (cf Smolin's book) are not well-founded anymore, in my opinion. But I will freely admit to knowing very little about this subject.
As for whether or not string theory NEEDS to be background independant, I don't know. If this is the ONLY objection to string theory, then I don't think it is a particularly well-founded one. I have never understood how to understand ``space-time'' in string theory. We are all comfortable saying that GR is a classical theory, an effective description of gravity at long distances, and that the geometry of space-time governs how we understand the gravitational dynamics. But this is not true on the quantum level, as I understand it. On the quantum level in string theory, we have gravitons, not geometry. So I don't really know what background independance MEANS for string theory.
If AlphaNumeric or pbrane is following, I would like to hear your opinions on this, as mine are probably wrong
The S matrix is the operator which governs the evolution of initial states to final states in a quantum field theory.
Ok, here goes. This is public, so you all will know how terrible a mathematician I am.
Let there be a vector space $$I$$ consisting of vectors $$\left| i \right\rangle$$ which describe the initial states of some system. There is a dual vector space, $$F$$, consisting of vectors $$\left\langle f \right|$$ which describe the possible final states of a physical system. There is a good inner product and basis, etc.
If we wish to find the probability that some initial state $$\left| i' \right\rangle$$ evolves into some final state $$\left\langle f' \right|$$ we take the inner product of these two states with the S matrix:
$$\left|\left\langle f'\right| S \left| i'\right\rangle\right|^2 = P_{i' \rightarrow f'} $$
I don't know if I've made this more clear or not...
I'm lost, but I don't blame Ben, for this, I blame Dirac!
Ben: You state that the operator $$S:I \to F$$ is a map from a vector space to its dual, fair enough - this is an isomorphism, right?
But earlier you wrote the form $$\left\langle f'|S|i' \rangle $$. Dirac gives us no information about the vector argument of S. By this form, it must be either $$|i' \rangle $$ or $$|f' \rangle $$. Or is $$\left\langle f'|S|i' \rangle $$ a bilinear form e.g. an inner product?
Even so, I do not understand the notation: if $$ S: I \to F$$, what is the justification for writing $$\left\langle f'|S|i' \rangle $$, which implies that $$|i' \rangle \in I$$, which is contrary to your opening assertion that $$|i \rangle \in I$$.
How can this be? What elements of $$I$$ is $$S$$ operating upon; is it $$|i \rangle$$ or $$ |i' \rangle$$? How are these guys related?
Or have I wildly missed the point?
Ben: You state that the operator $$S:I \to F$$ is a map from a vector space to its dual, fair enough - this is an isomorphism, right?
But earlier you wrote the form $$\left\langle f'|S|i' \rangle $$. Dirac gives us no information about the vector argument of S. By this form, it must be either $$|i' \rangle $$ or $$|f' \rangle $$. Or is $$\left\langle f'|S|i' \rangle $$ a bilinear form e.g. an inner product?
Even so, I do not understand the notation: if $$ S: I \to F$$, what is the justification for writing $$\left\langle f'|S|i' \rangle $$, which implies that $$|i' \rangle \in I$$, which is contrary to your opening assertion that $$|i \rangle \in I$$.
How can this be? What elements of $$I$$ is $$S$$ operating upon; is it $$|i \rangle$$ or $$ |i' \rangle$$? How are these guys related?
Or have I wildly missed the point?
I think S maps I to I because then you can make the duality pairing <f,Si>. Anyway, it is not important in case of Hilbert spaces since F is isometric to I via the Riesz representation theorem.
Ahh I should have made my notation more clear. I just wanted to be clear that i' was some other vector (like i) living in the space I. I feel like a child wearing my fathers boots when talking about math like this.
You are correct. I should write
$$\left\langle f|S|i \rangle$$.
Apologies all around.
Also, I think temur is correct in this point: