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Take the commutator acting on a function $f$. Then \begin{split} [ P_i , P_j ] f &= [ - i \partial_i - q A_i , - i \partial_j - q A_j ]f \\ &= ( i \partial_i + q A_i )( i \partial_j + q A_j ) f -( i \partial_j + q A_j ) ( i \partial_i + q A_i ) f \\ &= - \partial_i \partial_j + i q A_i \partial_j \, f + i q \partial_i ( ...

2

Why $m^2$ in front of $\phi^2$ and why is $m$ the mass? Fist of all, from dimensional analysis the prefactor to the $\phi^2$ term in the Lagrangian must have mass-dimension$^1$ $2$ in $3+1$ dimensions since the Lagrangian has mass-dimension $4$ and $\phi$ has mass-dimension $1$. This just tells us that we can write the term as $m^2\phi^2$ where $m$ is ...

1

Comments to the question (v8): Let us here for simplicity consider point mechanics. The generalization to field theory is straightforward and left to the reader. Given an (off-shell) action functional $$\tag{1} I[q]~=~\int_{t_i}^{t_f} \! dt~L,$$ it seems that OP in the first half of his post mainly confirms that the functional/variational derivative ...

1

Quantum fields cannot be turned on or off. The field itself exists for all time and space. It is possible to excite various modes of a quantum field at various spacetime points. These field excitations are interpreted as particles. When no excitations are present (i.e. no particles are present) the quantum field is in the vacuum state. Particles do not act ...

1

There is nothing wrong here. Choosing non-linear gauge conditions leads to uncanonical forms of the action. Generically, nothing ensures that for arbitary gauge conditions $F(A) = 0$ we will obtain some sort special form of the gauge fixed action. If one picks "odd" forms for the gauge fixing condition, then the resulting ghost action will contain unusual ...

1

A n-particle reducible diagram is a diagram that can be cut into two pieces if one cuts n or less lines. Conversely, a n-particle irreducible (n-PI) diagram cannot be cut into two pieces if one cuts n lines. The sunset diagram is 3-particle reducible, since it has 3 internal lines, but it is both 1-PI and 2-PI, and contributes to the self-energy (which ...

1

In the following calculation, I ignore some coefficients. According to $J(x)=\int d^4 k_1 e^{ik_1 x}$ , $J(y)=\int d^4 k_2 e^{ik_2 y}$ and $D(x-y)=\int d^4k \frac{e^{ik(x-y)}}{k^2-m^2+i\epsilon}$ We have $$W(J) = \int d^4x d^4y d^4 k d^4 k_1 d^4 k_1 J(k_1)e^{ik_1 x} J(k_2)e^{ik_2 x} \frac{e^{ik(x-y)}}{k^2-m^2+i\epsilon}$$ W(J)=\int d^4x d^4y d^4 k ...

1

The fields of a supersymmetric theory form a representation of the super Poincare algebra. When this representation is restricted to a specific value of the mass operator $P^{\mu}P_{\mu} = m^2$, the representation is called an on shell representation multiplet. On shell representations are characterized by the equality of the number of bosonic and ...

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