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1

Good luck. To check the cancellation for particular groups like $E_8\times E_8$ and $SO(32)$, you will indeed have to get through similar group-theoretical tasks. Similar trace formula for the traces of $E_8$ transformations are especially yummy, including the factor of $1/30$. The orthogonal case is easier even if one is not an intimate friend of all ...

3

$X_i,Y_i,Z_i$ are three Pauli matrices acting on the $i$-th qubit where $i=1,2,3,4,5,6,7,8,9$ labels the qubit. In equation 4.1, the state is a superposition of tensor product of three states similar to $|000\rangle$. The latter is a state of three qubits, so if one takes the third power, it is a state of $3\times 3 = 9$ qubits. $X_1$ differs from $X_8$ by ...

0

According to the Wikipedia article on String theory, At sufficiently high energies, the string-like nature of particles would become obvious. There should be heavier copies of all particles, corresponding to higher vibrational harmonics of the string. It is not clear how high these energies are. In most conventional string models, they would be close to ...

2

So my question is, is Fourier Analysis essentially what String Theory is? Briefly, no. String theory "is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings." Fourier analysis "is the study of the way general functions may be represented or approximated by sums of ...

0

It seems that this question is asked because of difficulties in trying to visualize higher dimensions. Fig.1: The full manifold is here $\mathbb{R}^2\times S^2$. I) Let us therefore for simplicity assume that the physical universe is just a 2D cylinder$^\dagger$ surface $\mathbb{R}\times S^1$. Imagine that there are only one large (uncompact) ...

0

Let me try to rephrase what you're asking. Suppose we have the usual spatial dimensions $x$, $y$ and $z$, and a compact spatial dimension $w$. Then can we have two particles simultaneously at positions: $$P_1 = (x, y, z, w)$$ and $$P_2 = (x, y, z, w + \delta w)$$ In other words the particles are at exactly the same position in the normal coordinates ...

1

Jerry Schirmer's answer to "Why are extra dimensions necessary?" seems (as far as I can tell) to give a good explanation of why compact dimensions are compact, from a QM point of view. I think it is more or less equivalent to the GR idea that mass/energy creates space and that the energy needed to create big space in higher dimensions is so much that it just ...

0

The question "why" is always a bit tricky. Elsewhere on PSE there is a good answer from Jerry Schirmer (Why are extra dimensions necessary?), it explains that other dimensions are compact because of the high energy requirement to make them grow bigger (my words). This seems to be a variation on the principle in GR that space only exists because of the ...

1

The conformal compactification is supposed to belong to the projective space so we still identify points along the rays (equivalence classes under scaling) $$(u,v,x_i)\sim \lambda (u,v,x_i), \quad \lambda \neq 0$$ Then there is the quadric equation you wrote down – an equation that respects the identification above – so both added variables $u,v$ are pretty ...

1

In principle it should not. The problem with the Wick rotation is that what you are doing is embedding the Lorentzian manifold in a complex manifold of which it is a slice, and then looking for a different slice with Riemannian signature. In general there is no such Riemmanian slice, and even if it does exist it need not be unique. There is an old paper ...

4

Yes, one may discuss ordinary quantum mechanics based on point-like particles using the 0+1-dimensional quantum field theory (i.e. quantum mechanics) defined on the world line. This is actually how several textbooks of string theory, including Polchinski's textbook, begin. The Green's functions $G(x,y)$ may be calculated as a traditional "Feynman sum over ...

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