Megascopic Quantum Phenomena
319
∂
∂ x μ
∂ L
∂
∂ψ
∂ x μ
−
∂ L
∂ψ
= 0
(10.2)
and the Hamiltonian density H, obtained from the Lagrangian density L by means
of the Legendre transformation
H =
i
˙
ψ i
∂ L
∂ ˙
ψ i
− L
(10.3)
Comay [88] made four main contentions against the quantum theoretical consistency
of the Higgs mechanism. Using units, è = c = 1, the Higgs boson Lagrangian density
writes in the following compact form:
L Higgs = φ
+
,μ φ ,ν g
μν
+ m
2
φ
+
φ + OT
(10.4)
where φ is the scalar function of the Higgs boson, m denotes the Higgs mass and OT
denotes other terms (cubic etc.).
The first Comay argument avers: The action S in Eq. (10.1) is a dimensionless
Lorentz scalar, while the Lagrangian density L in (10.1) and (10.4) is a Lorentz scalar
whose dimension is [L
–4 ]. Hence, the first term of (10.4) proves that the dimension
of the Higgs function φ is [L
–1 ]. Since the density of a quantum particle is given in
terms of its wave function
ρ = |ψ|
2
(10.5)
the Schrödinger density satisfies the continuity equation, with the dimension of the
wave function is [L
–3/2 ]. The difference in dimensions between the Higgs and the
Schrödinger functions indicates that Higgs theory does not have a non-relativistic
limit. This is an inconsistency in the broader sense of Bohr’s correspondence
principle.
The second Comay argument reads: The Higgs function φ is a scalar function
whose dimension is [L
–1 ]. Hence, the product φ
+
φ is a scalar with dimension [L
–2 ].
On the other hand, the density is the zeroth component of a 4-vector whose dimension
is [L
–3 ]. Hence, the product φ
+
φ does not represent the density of the Higgs boson.
This is another inconsistency of the Higgs theory with the Bohr’s correspondence
principle.
The third Comay argument says: Examining the Euler-Lagrange equation of the
Higgs boson, one finds, applying Eq. (10.2) to Higgs Lagrangian density (10.4), the
following equation for φ:
φ − m
2
φ + OT = 0
(10.6)
The Euler-Lagrange equation of the particle’s Lagrangian density must agree with
the fundamental quantum mechanical equation, the Schrödinger equation, which is
319
∂
∂ x μ
∂ L
∂
∂ψ
∂ x μ
−
∂ L
∂ψ
= 0
(10.2)
and the Hamiltonian density H, obtained from the Lagrangian density L by means
of the Legendre transformation
H =
i
˙
ψ i
∂ L
∂ ˙
ψ i
− L
(10.3)
Comay [88] made four main contentions against the quantum theoretical consistency
of the Higgs mechanism. Using units, è = c = 1, the Higgs boson Lagrangian density
writes in the following compact form:
L Higgs = φ
+
,μ φ ,ν g
μν
+ m
2
φ
+
φ + OT
(10.4)
where φ is the scalar function of the Higgs boson, m denotes the Higgs mass and OT
denotes other terms (cubic etc.).
The first Comay argument avers: The action S in Eq. (10.1) is a dimensionless
Lorentz scalar, while the Lagrangian density L in (10.1) and (10.4) is a Lorentz scalar
whose dimension is [L
–4 ]. Hence, the first term of (10.4) proves that the dimension
of the Higgs function φ is [L
–1 ]. Since the density of a quantum particle is given in
terms of its wave function
ρ = |ψ|
2
(10.5)
the Schrödinger density satisfies the continuity equation, with the dimension of the
wave function is [L
–3/2 ]. The difference in dimensions between the Higgs and the
Schrödinger functions indicates that Higgs theory does not have a non-relativistic
limit. This is an inconsistency in the broader sense of Bohr’s correspondence
principle.
The second Comay argument reads: The Higgs function φ is a scalar function
whose dimension is [L
–1 ]. Hence, the product φ
+
φ is a scalar with dimension [L
–2 ].
On the other hand, the density is the zeroth component of a 4-vector whose dimension
is [L
–3 ]. Hence, the product φ
+
φ does not represent the density of the Higgs boson.
This is another inconsistency of the Higgs theory with the Bohr’s correspondence
principle.
The third Comay argument says: Examining the Euler-Lagrange equation of the
Higgs boson, one finds, applying Eq. (10.2) to Higgs Lagrangian density (10.4), the
following equation for φ:
φ − m
2
φ + OT = 0
(10.6)
The Euler-Lagrange equation of the particle’s Lagrangian density must agree with
the fundamental quantum mechanical equation, the Schrödinger equation, which is