(SR) String Theory Q and A

BenTheMan

Dr. of Physics, Prof. of Love
Valued Senior Member

Every rigorous theory should have well defined postulate list. Without postulate list defined every theory can claim virtually everything, it cannot be distinguished from other quantum field theories and as such it's untestable by such way. For example, we can find the postulates of relativity theory or quantum mechanics on the web.

Where we can found the postulates of string theory, which is developed over forty years already? Please consider, no vague trolling - just rigorous answers are supposed in this thread.

Postulates of string theory:

1.) The fundamental object is a string.

This is the only postulate that I know of. Everything follows from this assumption, and the requirement that the thoery be non-anomalous.

Does the string fulfill the wave equation or some other equations? Isn't the string theory QM and SR dependent? At this moment we have dozen of additional postulates. The assumption of higher (10, 11 or more) dimensions is the another one.

• The current prevailing string theory, called M-Theory proposes an eleven-dimensional space that consists of objects with multiple dimensions called p-branes. One type of p-brane is the d-brane, which can be related to the end points of the strings.
• Bosonic string theory is operating in 26 dimensions. It considers only bosons, no fermions means only forces, no matter, with both open and closed strings. It considers a particle with imaginary mass, i.e. the tachyon.
• String I works with both open and closed strings, no tachyon, group symmetry is SO(32)
• String IIA works with closed strings only, uses no tachyons, massless fermions spin both ways (nonchiral)
• String IIB works with closed strings only, uses no tachyons, massless fermions only spin one way (chiral)
• Heterotic string HO theory works with closed strings only, uses no tachyons, heterotic, meaning right moving and left moving strings differ, it uses group symmetry SO(32)
• Heterotic string HE theory works with closed strings only, uses no tachyons, meaning right moving and left moving strings differ, group symmetry is E8 x E8 (it uses two copies of the E8 lattice to hide the extra dimensions)

Yes.
These things can be seen to follow from string theory.

Also, the number of space-time dimensions is not an assumption of string theory. This is very important. Unlike in GR, the number of dimensions in string theory is a derived quantity.

Bosonic string theory is operating in 26 dimensions. It considers only bosons, no fermions means only forces, no matter, with both open and closed strings. It considers a particle with imaginary mass, i.e. the tachyon.

Bosonic string theory is a TOY theory, that is, it cannot describe nature because it has no fermions. So the fact that it is inconsistent (i.e. presence of a tachyon) is not a problem.

zephir said:
Every rigorous theory should have well defined postulate list. Without postulate list defined every theory can claim virtually everything, it cannot be distinguished from other quantum field theories and as such it's untestable by such way. For example, we can find the postulates of relativity theory or quantum mechanics on the web.

Where we can found the postulates of string theory, which is developed over forty years already? Please consider, no vague trolling - just rigorous answers are supposed in this thread.

Also, string theory requires that the S matrix be unitary.

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?

Er.... have I missed something? What is "the S matrix"?

Er.... have I missed something? What is "the S matrix"?

initial state and final state of interaction of particles.

Sorry to be dim, but I am no wiser. Do you think you could make this more comprehensible to ignorants like me? Like, a slightly fuller explanation?

I think f=Si where f is the final state and i is the initial state.

Ben,
I thought the S matrix was by definition unitary regardless of string theory. (Disclaimer: Unfortunately, I am no mathematician)

Well, perhaps I should say that the evolution is goverened by a unitary S matrix.

Possibly related general question:
Does string theory need to be background independent? Has string FIELD theory been completely dropped?

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

Sorry to be dim, but I am no wiser. Do you think you could make this more comprehensible to ignorants like me? Like, a slightly fuller explanation?

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...

So S is an operator S : I -> I.

S maps I to F.

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?

S maps I to F.

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.

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?

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:

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.