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The Hamiltonian for a scalar field contains the term $$\int d^3x m^2 \phi(x) \phi(x)$$, does it changs to the following form? $$\int d^3x' {m'}^2 \phi'(x') \phi'(x')=\int d^3x' \gamma^2{m}^2 \phi(x) \phi(x)$$? As it is well known for a scalar field: $$\phi'(x')=\phi(x)$$ .

my questions stemmed from reading the article in Physica E. Vol. 86, 10-16. (https://www.sciencedirect.com/science/article/pii/S1386947716311365) Why does the graphene Fermi velocity ##v_F## appear in Eq.(11) in this article,? Eq.(11) is as follows: $$ \frac{\partial \Omega_p(z,t)}{\partial...

Thank you very much, I will read the relevant references according to your guidance.

thank you for your comment, I thought the author's meaning is ##g_{xx}## depends only on x; ##g_{yy}## depends only on y; etc.

I encountered a problem in reading Phys.Lett.B Vol.755, 367-370 (2016). I cannot derive Eq.(7), the following snapshot is the paper and my oen derivation, I cannot repeat Eq.(7) in the paper. ##g^{\mu\nu}## is diagonal metric tensor and##g^{\mu\mu}## is the function of ##\mu## only...

Thank you! Yes, you are right.

Thank you very much for your quick reply, as you know the unnumbered equation in page 194 closely below the sentence "it is necessary and sufficient that" is obtained from 5.1.6 and 5.1.11. the left hand side of this unnumbered equation can be derived from 5.1.6 and above here, while the right...

According to (5.1.6) $$U_0(\Lambda,a)\psi_\ell^+(x)U^{-1}_0(\Lambda,a)=\sum\limits_{\ell \bar{\ell}}D_{ \ell \bar{\ell} }(\Lambda^{-1})\psi^+_{\bar{\ell}}(\Lambda x+a).$$ (5.1.6) According to definition 5.1.4: $$\psi^+_{\bar{\ell}}(\Lambda x+a)=\sum\limits_{\sigma n}\int d^3{\bf p }...

Thank you very much for your reply.

As is well known there is translation operator in position space, such that., $$\exp(i\hat{p}a)x\exp(-i\hat{p}a)=x+a.$$ While in momentum space, can we have analog of the above mentioned translation operator? i.e., momentum shift operator? $$\exp(-i\hat{x}q)p\exp(i\hat{x}q)=p+q.$$ If so, why...

Thank you very much , with my best wishes.

Thank you very much for your help, I mean is the tight binding wavefunction (lattice coefficient) at atom A (or B) two-component spinor? If so, how the spinorial lower component propogate? If not, how tight binding wavefunction match the continuum model? In continuum model, the wavefunction...

In the framework of tight binding approximation, does the wavefunction for atom A (or B) has two spinorial components(2 components) in "real space"? If so how does this spinorial component propagate in the graphene?

It is well known that the 2D free electron gas fermi momentum can be expressed as follows, k_F=\left(2\pi n\right)^{1/2} where n is the electron surface density. Assuming this 2D electron system can be considered as 2-D tight-binding square lattice whose eigenergy can be written as...