Fermi's theory of beta decay - Does Fermi's Hamiltonian have the wrong transformation properties?
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by Fermi in the 30's, and I found an inconsistency between the transformation properties that he claims for his Hamiltonian and the transformation properties that Weisskopf and Blatt give about the very same Hamiltonian in the book "Theoretical Nuclear Physics". You'll find Fermi's original german article translated here (see in particular Section III).
Now, Fermi claims his Hamiltonian to be the time-component of a four-vector interaction, similar in form to the leptonic charged current that couples to the W boson in the standard model (it basically lacks, of course, the pseudovector interaction time component). He starts by enumerating four bilinear combinations of the components of the electron's (ψ) and antineutrino's (ϕ) Dirac spinors, which altogether transform as a four-vector:
A0=−ψ1ϕ2+ψ2ϕ1+ψ3ϕ4−ψ4ϕ3A1=ψ1ϕ3−ψ2ϕ4−ψ3ϕ1+ψ4ϕ2A2=iψ1ϕ3+iψ2ϕ4−iψ3ϕ1−iψ4ϕ2A3=−ψ1ϕ4−ψ2ϕ3+ψ3ϕ2+ψ4ϕ1
(see eq. 11 in Fermi's article; given the time of writing, the spinors should be taken to satisfy Dirac's equation in the Dirac's basis). Supposing that the decaying nucleon's velocity is much less than the speed of light, the Hamiltonian can be taken to contain only the time component of the vector (in analogy to the electromagnetic case, the spatial ones should couple weakly to the nucleon's wavefunction). So he explicitly writes:
H=g[ψ†δϕ∗ Q†+ψTδϕ Q]
(see eq. 13) where the Hamiltonian is taken to act between the initial and final wavefunction of the nucleon, the operator Q replaces a proton with a neutron (and Q† does the opposite), δ is the matrix
δ=⎛⎝⎜⎜⎜0100−1000000−10010⎞⎠⎟⎟⎟
and ψ and ϕ this time are the Dirac's bispinors multiplied by the creation/annihilation operators of the electron and neutrino, to be evaluated at the nucleon's position. Notice two things: first of all, as the proton's and neutron's creation/annihilation operators do not appear explicitly in the Hamiltonian, Fermi's formalism treats the heavy and the light particles on a different footing (but this is only a matter of writing) - moreover the familiar ∫d3x integral must be taken only when calculating H's matrix elements; second of all, Fermi's conventions about which neutrino (i.e. the neutrino or the antineutrino) should appear together with the electron/positron are opposite to the modern formalism: today we would call ϕ∗ the neutrino's field instead of the antineutrino's creation operator (of course the two could be the same, but the distinction is useful if one wants to compare Fermi's Hamiltonian to the leptonic charged weak current).
On the other hand, in Blatt and Weisskopf's beautiful analysis of the possible interaction terms that can appear in the nucleus's Hamiltonian (see Ch. XIII, Sec. 5.C), the very same term, whose electron/neutrino portion for β− decays is expressed in Fermi's notation as
FS=ψ†δϕ∗
is said to be a scalar (S) under the Poincaré group, while the time component of the four-vector interaction is given as
FV0=ψ†βδϕ∗
where β is Dirac's beta matrix, i.e. (in the Dirac's basis)
Now, Fermi claims his Hamiltonian to be the time-component of a four-vector interaction, similar in form to the leptonic charged current that couples to the W boson in the standard model (it basically lacks, of course, the pseudovector interaction time component). He starts by enumerating four bilinear combinations of the components of the electron's (ψ) and antineutrino's (ϕ) Dirac spinors, which altogether transform as a four-vector:
A0=−ψ1ϕ2+ψ2ϕ1+ψ3ϕ4−ψ4ϕ3A1=ψ1ϕ3−ψ2ϕ4−ψ3ϕ1+ψ4ϕ2A2=iψ1ϕ3+iψ2ϕ4−iψ3ϕ1−iψ4ϕ2A3=−ψ1ϕ4−ψ2ϕ3+ψ3ϕ2+ψ4ϕ1
(see eq. 11 in Fermi's article; given the time of writing, the spinors should be taken to satisfy Dirac's equation in the Dirac's basis). Supposing that the decaying nucleon's velocity is much less than the speed of light, the Hamiltonian can be taken to contain only the time component of the vector (in analogy to the electromagnetic case, the spatial ones should couple weakly to the nucleon's wavefunction). So he explicitly writes:
H=g[ψ†δϕ∗ Q†+ψTδϕ Q]
(see eq. 13) where the Hamiltonian is taken to act between the initial and final wavefunction of the nucleon, the operator Q replaces a proton with a neutron (and Q† does the opposite), δ is the matrix
δ=⎛⎝⎜⎜⎜0100−1000000−10010⎞⎠⎟⎟⎟
and ψ and ϕ this time are the Dirac's bispinors multiplied by the creation/annihilation operators of the electron and neutrino, to be evaluated at the nucleon's position. Notice two things: first of all, as the proton's and neutron's creation/annihilation operators do not appear explicitly in the Hamiltonian, Fermi's formalism treats the heavy and the light particles on a different footing (but this is only a matter of writing) - moreover the familiar ∫d3x integral must be taken only when calculating H's matrix elements; second of all, Fermi's conventions about which neutrino (i.e. the neutrino or the antineutrino) should appear together with the electron/positron are opposite to the modern formalism: today we would call ϕ∗ the neutrino's field instead of the antineutrino's creation operator (of course the two could be the same, but the distinction is useful if one wants to compare Fermi's Hamiltonian to the leptonic charged weak current).
On the other hand, in Blatt and Weisskopf's beautiful analysis of the possible interaction terms that can appear in the nucleus's Hamiltonian (see Ch. XIII, Sec. 5.C), the very same term, whose electron/neutrino portion for β− decays is expressed in Fermi's notation as
FS=ψ†δϕ∗
is said to be a scalar (S) under the Poincaré group, while the time component of the four-vector interaction is given as
FV0=ψ†βδϕ∗
where β is Dirac's beta matrix, i.e. (in the Dirac's basis)
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I'm studying the theory of beta decays as proposed by Fermi in the 30's, and I found an inconsistency between the transformation properties that he claims for his Hamiltonian and the transformation properties that Weisskopf and Blatt give about the very same Hamiltonian in the book "Theoretical Nuclear Physics".
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