I'm trying to show that \int d^3x \,x^\mu \left(\partial_\mu \partial_0-g_{\mu 0} \partial^2 \right)\phi^2(x)=0 . This term represents an addition to a component of the energy-momentum tensor \theta_{\mu 0} of a scalar field and I want to show that this does not change the dilation operator...
It seems to me that in a path integral, since you are integrating over all field configurations, that going into Euclidean space is not valid because some field configurations will give poles in the integrand of your action, and when the integrand has poles you can't make the rotations required...
Without interaction, then in an expression like \langle p |\phi^4(x)|p\rangle , the bra and the ket are the same state: <p|p>=1.
However, with interaction, <p|p>≠1, since the bra is an Out-state, and the ket is an In-state.
It seems if you calculate \langle p |\phi^4(x)|p\rangle with...
Suppose you want the 1-particle matrix elements of an operator in QFT, e.g.
\langle p' |\phi^4(x)|p\rangle
It seems you would calculate this perturbatively by first Fourier transforming the x-variable to q, assuming an incoming particle with momentum p, an outgoing particle with momentum p'...
Suppose you have a λφ4 theory. Books only seem to calculate counter-terms for 2-pt and 4-pt functions.
But what about 3 particles scattering into 3 particles? Do the counter-terms determined by renormalizing the 2-pt and 4-pt functions cancel divergences in 3x3 scattering?
For example, take...
Are a field and its complex conjugate independent? It seems like they're not, as one is the complex conjugate of the other, so if you have one, you know the other.
However, it seems in path integrals, you integrate over the field and its conjugate, so they can take on values that are not the...
In a high energy physics experiment, do you ever scatter more than 2 particles at a time?
Suppose you scatter 4 incoming particles and get 4 outgoing particles. Do you only look at connected diagrams with 8 external lines? Or do you also have to take into account the product of disconnected...
Can Jacobi Identity and Leibniz derivation alone tell you what group it is?
For example, take the SO(3) commutation relations. If you change [Jx,Jy]=iJz to [Jx,Jy]=-iJz and leave all other commutators the same, then I think you get something like SO(2,1) rather than SO(3). SO(2,1) would still...
Is there a way to determine the group from the commutation relations?
For example, the commutation relations:
[J_x,J_y]=i\sqrt{2} J_z
[J_y,J_z]=\frac{i}{\sqrt{2}} J_x
[J_z,J_x]=i\sqrt{2} J_y
is actually SO(3), as can be seen by redefining J'_x =\frac{1}{\sqrt{2}} J_x : then J'_x , J_y and...
Thanks. If I have a correlation function, isn't it usually infinity even if all fields in the correlator are at different spacetime points? The exception seems to be a free field theory where everything is finite: therefore it seems infinitys come from structure of hamiltonian, and not the...
Is it okay to define a local operator as an operator whose matrix elements in position space is a finite sum of delta functions and derivatives of delta functions with constant coefficients?
Suppose your operator is M, and the matrix element between two position states is <x|M|y>=M(x,y).
It...
Don't counter-terms have one more power of the coupling in them than bare terms? For example, for λΦ4, I worked out the the coupling has dimensions [\lambda]=\frac{1}{\hbar c}.
So if we write \lambda=k*\frac{1}{\hbar c}, where k is a pure number, then whether your higher order perturbative...
In the saddle point evaluation of the path integral, at tree level, you plug in the classical solution of the field into the integrand. However, when determining the classical solution, we ignore counterterms. The counterterms only show up to renormalize divergences after a saddle point...
Suppose you have a quantized Dirac theory. Can you get away with just describing the electron's degree of freedom by ignoring the creation and annihilation operators for the positron in your free fields (which are used in the interaction picture)? Basically set the positron creation and...
Suppose you want to evaluate:
$$\langle \Omega | T\phi(x) \phi(y) | \Omega \rangle$$
where \Omega is the ground state of the full Hamiltonian, and the fields are in the Heisenberg representation. Assume x_0 > y_0 for now. Then it's straightforward to show that
$$\langle \Omega | T\phi(x)...