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In Srednicki's proof of the spin-statistics theorem for scalar fields ("Quantum Field Theory", section 4), he considers interaction terms, added to the free Hamiltonian, that are Hermitian function...
#2: Post edited
by
Mithrandir24601
·
2021-05-06T18:42:07Z (over 3 years ago)
Added link to Srednicki's page of textbook
In Srednicki's proof of the spin-statistics theorem for scalar fields ("Quantum Field Theory", section 4), he considers interaction terms, added to the free Hamiltonian, that are Hermitian functions of$$\phi^+(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{ikx}a(\mathbf{k})$$- and
$$\phi^-(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{-ikx}a^\dagger(\mathbf{k}).$$- He then goes on to show that we will only get Lorentz-invariant transition amplitudes if we are dealing with the theory for a real scalar field obeying Bose-Einstein statistics.
**Question:** Why do we only need to consider interaction terms that are functions of $\psi^+$ and $\psi^-$? Couldn't there be other possible interaction terms? For example, $H_1=a^\dagger(\mathbf{k})+a(\mathbf{k})$?
- In [Srednicki's](https://web.physics.ucsb.edu/~mark/qft.html) proof of the spin-statistics theorem for scalar fields ("Quantum Field Theory", section 4), he considers interaction terms, added to the free Hamiltonian, that are Hermitian functions of
- $$\varphi^+(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{ikx}a(\mathbf{k})$$
- and
- $$\varphi^-(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{-ikx}a^\dagger(\mathbf{k}).$$
- He then goes on to show that we will only get Lorentz-invariant transition amplitudes if we are dealing with the theory for a real scalar field obeying Bose-Einstein statistics.
- **Question:** Why do we only need to consider interaction terms that are functions of $\varphi^+$ and $\varphi^-$? Couldn't there be other possible interaction terms? For example, $H_1=a^\dagger(\mathbf{k})+a(\mathbf{k})$?
#1: Initial revision
by
Technically Natural
·
2021-04-02T18:48:46Z (over 3 years ago)
Interaction terms in Srednicki's proof of spin-statistics theorem
In Srednicki's proof of the spin-statistics theorem for scalar fields ("Quantum Field Theory", section 4), he considers interaction terms, added to the free Hamiltonian, that are Hermitian functions of
$$\phi^+(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{ikx}a(\mathbf{k})$$
and
$$\phi^-(x)=\int\frac{d^3k}{(2\pi)^32\omega}e^{-ikx}a^\dagger(\mathbf{k}).$$
He then goes on to show that we will only get Lorentz-invariant transition amplitudes if we are dealing with the theory for a real scalar field obeying Bose-Einstein statistics.
**Question:** Why do we only need to consider interaction terms that are functions of $\psi^+$ and $\psi^-$? Couldn't there be other possible interaction terms? For example, $H_1=a^\dagger(\mathbf{k})+a(\mathbf{k})$?