Questions from chapter 2 of Zwiebach's A First Course In String Theory

[sciencegem]

Hi,

I have a couple questions from chapter 2 of Zwiebach's A First Course In String Theory, second ed.

1) Zwiebach says [in the context of compact extra dimensions] "it seems very difficult, if not altogether impossible, to construct a consistent theory with more than one time dimension". Why is that? The best explanation my simple brain on Kruskal coordinates came up with is going faster than the speed of light is analogous to utilizing an extra time dimension to some extent, and (in the unlikely scenario I'm correct) if the converse were also true, then an extra time dimension would equate surpassing the speed of light. That being said, I can imagine constructing perfectly obedient metrics with extra negative signatures as easily as I can do so with positive signatures (is my terminology there correct?).

2) Zwiebach writes Schrodinger's equation as

E = -(\hbar²/2m)(1/ψ(x))(d²ψ(x)/dx²) -(\hbar²/2m)(1/ø(y))(d²ø(y)/dy²)

and derives

ψ_k(x) = c_ksin(k*π/a)

ø_l(y) = a_lsin(l*y/R) + b_lcos(l*y/R)

as solutions, and

E_k,l = (\hbar²/2m)[(k*π/a)² + (l/R)²]

as the energy eigenvalues.

How did he derive those? The trig functions kinda came out of nowhere for me, and I'm unfamiliar with the concept of taking eigenvalues from (what appears to my untrained eye to be) a regular equation (i.e. not a matrix). Sorry for my stupidity here, I gather I probably don't have the appropriate prerequisites for the book, so if anyone has any resource suggestions to help me gain the appropriate background I would be grateful for the advice. Any hints on how I might resolve my questions are appreciated. Also, if anyone knows where I can access the solutions to the problems for the text that would be just awesome. I imagine it will be very difficult trying to glean adequate intuition from this book if all my answers are wrong but in my ignorance I assume they are correct.

Thanks to anyone who read all that, like I said hints appreciated.

 

[AndyW]

Hi there,

Thank you for your questions regarding Zwiebach's A First Course In String Theory. I am happy to provide some insight and clarification on these topics.

1) The reason why it is difficult, if not impossible, to construct a consistent theory with more than one time dimension is due to the concept of causality. In physics, causality means that the cause of an event must always precede its effect. This is a fundamental principle that is observed in all physical phenomena. In a theory with more than one time dimension, it becomes impossible to define a clear notion of causality. This leads to inconsistencies and contradictions within the theory. Additionally, the mathematical framework for such a theory becomes very complicated and difficult to work with. Therefore, most physicists do not consider theories with more than one time dimension to be viable.

Your analogy with Kruskal coordinates is interesting, but it is important to note that these coordinates are purely mathematical tools used to simplify the description of black holes in general relativity. They do not necessarily have a physical interpretation. Furthermore, the concept of faster-than-light travel is not well-defined in string theory, so it is not appropriate to use it as a basis for understanding the limitations of multiple time dimensions.

2) The solutions for the Schrodinger equation that Zwiebach presents are derived using a combination of mathematical techniques and physical principles. The trigonometric functions are a result of solving the differential equations that arise from the Schrodinger equation. The concept of eigenvalues and eigenvectors is a fundamental tool in quantum mechanics, and it is used to describe the energy states of a system. In this case, the energy eigenvalues are obtained by solving the differential equations for the wave functions.

To better understand these concepts, I would recommend studying some introductory texts on quantum mechanics and differential equations. Some good resources are "Introduction to Quantum Mechanics" by David J. Griffiths and "Elementary Differential Equations and Boundary Value Problems" by William E. Boyce and Richard C. DiPrima. Additionally, it may be helpful to review some linear algebra, as the concept of eigenvalues is closely related to the diagonalization of matrices.

Unfortunately, I am not aware of any publicly available solutions for the problems in Zwiebach's book. However, I would recommend working through the problems to gain a better understanding of the material.

I hope this helps to answer your questions. Please let me know if you have any further inquiries.

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