Recent content by aesir

Yes, those are some of the coefficients of the mass matrix, to understand also the diagonal part try to expand this product: \left(\begin{array}{ll} \pi^0 & \eta\end{array}\right)\left(\begin{array}{ll} m_1^2 & m_{12} \\ m_{12} & m_2^2\end{array}\right)\left(\begin{array}{l} \pi^0 \\...

I have not checked the maths, but: - for a complex field the mass term is proportional to \phi^\dagger\phi, if expanded it in two hermitian fields it is \phi_1^2+\phi_2^2 - \eta and \pi_0 have a mass matrix in the Lagrangian you quoted, physical particles are eigenvalues of mass, try to...

In fact we don't care! The only important thing is that somebody (= 't Hooft) as proved that different procedures give the same results, and there exists a procedure that preserves those symmetries. So we know that the result has Lorentz and gauge symmetries and you can chose any procedure to...

If the problem is 2 dimensional, then you can diagonalize the matrix and get an analytical result: http://www.wolframalpha.com/input/?i=diagonalize+{{a%2Cb}%2C{b%2Cc}}

It is much simpler in terms of exponentials, for example write cos(x)=(e^{ix}+e^{-ix})/2

No, the next value is \Gamma(1)=\lim_{x\to 0}x \Gamma(x): using the previous limit would give 0\times\infty which is undetermined. You are on the right track with the recurrence relation: can you find an expression for \Gamma(1+x) when x\to0?

I found a general demonstration in Courant & Hilbert "Methods of Mathematical Physics" Vol.I p 452 (you can read it from google books). But for one dimension it can be intuitively understood by elementary methods. Consider a real function which solves the equation for a generic energy E...

For one dimensional problems the eigenstates are ordered by their number of nodes (the points in which the wavefunctions is 0). So an odd wavefunction can not be the ground state.

You can extend the domain to infinity writing the function as the product of f and the characteristic function of the cube. Then by the convolution theorem the Fourier transform is the convolution of f_k and something like sinc(k). Don't know if you can go any further

It seems that you already understood that (0,0) is the problematic point. If you choose a symmetric domain about (0,0) you can calculate those integral analytically. Any other domain is just a small symmetric one plus something else.

I have not tried yet, but a stirling's approximation of {}_aC_n would give the correct limit n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n

Wikipedia calls it Sokhatsky-Weierstrass theorem

There is no mass operator in (non-relativistic) quantum mechanics. You could define one, but it turns out that every physical state of a single particle system must be an eigenvector of mass, technically we say that the mass is a super-selected observable. The reason for this...

It's not true in general. Counterexample: take the pauli matrices [\sigma_x,\sigma_y^2]=[\sigma_x,1]=0, but [\sigma_x,\sigma_y]=2i\sigma_z \neq0

The expression I wrote is the correct relativistic form of kinetic energy, since your speed is comparable to the speed of light (in the reference frame where the ship is at rest at t=0) you can not understand motion with Newtonian laws. \gamma is the Lorentz factor...