Boudet R. Quantum Mechanics in the Geometry of Space-Time: Elementary Theory
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Part II
The U(1) Gauge in the Complex
and Real Languages. Geometrical
Properties and Relation with the Spin
and the Energy of a Particle of Spin 1/2
Chapter 4
Geometrical Properties of the U(1) Gauge
Abstract A change of gauge and the condition of a gauge invariance, as well on
the wave function of a particle, as upon a potential vector acting on the particle, is
recalled f or the complex language and established for the real one.
Keywords U(1)
·SO
+
(1, 3) ·Finite ·Infinitesimal rotations ·Energy
4.1 The Definition of the Gauge and the Invariance
of a Change of Gauge in the U(1) Gauge
4.1.1 The U(1) Gauge in Complex Language
Let us denote by the Dirac spinor (defined by a column of four complex numbers
upon which the Dirac matrices can act) associated with a particle. A change of gauge
is defined by the transform
→
= U = U, U = e
iχ/2
, i =
√
−1,χ∈ R (4.1)
where, we recall
i = i (4.2)
The number χ may be fixed or dependent on the point x of M and in this cases the
change of gauge is to be said global or local.
4.1.2 The U(1) Gauge Invariance in Complex Language
Let us consider, associated to a particle submitted to a potential A ∈ M, an expression
in the form
R. Boudet, Quantum Mechanics in the Geometry of Space–Time,21
SpringerBriefs in Physics, DOI: 10.1007/978-3-642-19199-2_4,
© Roger Boudet 2011
22 4 Geometrical Properties of the U(1) Gauge
L
μ
= ∂
μ
i − gA
μ
, (4.3)
where is a Dirac spinor expressing the wave function of the particle and g a suitable
physical constant.
In the Dirac theory of the electron one has g = q/(c) where q =−e, e > 0, is
the charge of the electron and so gA
μ
has the dimension of the inverse of a length.
In a change of a local gauge like Eq. 4.1, L
μ
becomes
L
μ
=
∂
μ
i −
1
2
∂
μ
χ + gA
μ
U (4.4)
If L
μ
is chosen in such a way that L
μ
= L
μ
U, nothing is changed, except the
transform of into U, if A
μ
is changed into
A
μ
= A
μ
−
∂
μ
χ
2g
, A
= A −
∂χ
2g
∈ M,∂= e
μ
∂
μ
∈ M (4.5)
Such a change of the potential A does not affect the field F = ∂ ∧ A associated with
A, because ∂χ is a gradient.
4.1.3 A Paradox of the U(1) Gauge in Complex Language
The definition Eq. 4.1 of the U(1) gauge in complex language leads to a paradox.
Since i is considered as nothing else but the number
√
−1, a change of gauge must
be interpreted as related to some abstract property of the wave function , and, as a
consequence, the number χ has no geometrical meaning. But in a change of gauge
it must be considered with a geometrical meaning, because it appears in addition to
a potential which is a vector of the Minkowski space–time M, and so as a gradient
in M, that is a geometrical object.
Thus a geometrical interpretation of the gauge U(1) appears as a necessity.
4.2 The U(1) Gauge in Real Language
A geometrical interpretation of the U(1) gauge as a sub-group of SO
+
(1, 3) has been
implicitly contained in [1], explicitly and independently described in [2, 3]. But it
is unknown to most physicists and sometimes violently negated by some of them.
Their reason is related to the meaning of the number i in the theory of the electron
and their ignorance of the relation (4.6): since i = i they say that it is impossible
that U(1) may be interpreted as corresponding to a sub-group of SO
+
(1, 3).
4.2 The U(1) Gauge in Real Language 23
4.2.1 The Definition of the U(1) Gauge in Real Language
The Dirac wave function being expressed in the form Eq. 3.1 given by Hestenes [3],
Eq. 4.4 transforms Eq. 4.1 into
ψ → ψ
= ψU, U = e
e
2
e
1
χ/2
,χ∈ R (4.6)
where
e
e
2
e
1
χ/2
= cos(χ /2) + sin(χ/2)e
2
e
1
Note that ψ is to be multiplied on the right by U = e
e
2
e
1
χ/2
.
But here, what is called a change of gauge U(1) in complex language, corresponds
in STA to
U = e
e
2
e
1
χ/2
, R → R
= RU = Re
e
2
e
1
χ/2
(4.7)
which i nduces a rotation through an angle χ in the plane (n
2
, n
1
) :
n
2
= cos χ n
2
+ sin χ n
1
, n
1
=−sin χ n
2
+ cosχ n
1
(4.8(1))
with
R
e
0
R
−1
= e
0
, R
e
3
R
−1
= e
3
(4.8(2))
So the replacing of iχ/2bye
2
e
1
χ/2givestoχ the real geometrical meaning of
an angle.
Furthermore, since the plane (n
2
, n
1
) (in “up”, or (n
1
, n
2
) in “down”) is the plane
defined by the bivector spin of a particle of spin 1/2, one can give to the U(1) gauge
the following definition:
The U(1) gauge is the ring of the rotations upon itself of the plane defined by the
bivector spin of a particle of spin 1/2.
4.2.2 The U(1) Gauge Invariance in Real Language
The r eplacing of i =
√
−1 by the bivector e
2
∧ e
1
= e
2
e
1
in Eq. 4.3 gives
L
μ
= ∂
μ
ψe
2
e
1
− gA
μ
ψ, (4.9)
But here e
2
e
1
is necessarily to be written on the right of ψ, when one can
write i = i. Apart from this difference, the calculation is the same as in
Sect. 4.1.2 and gives also Eq. 4.5 but here the paradox evoked above disappears.
24 4 Geometrical Properties of the U(1) Gauge
A change of gauge through an angle χ implies
ω → ω
= ω − ∂χ ∈ M (4.10)
where
ω
μ
= ∂
μ
n
2
.n
1
=−∂
μ
n
1
.n
2
(4.11)
which expresses the infinitesimal rotation upon itself of the plane (n
2
, n
1
).
The i nvariance is achieved by the change of the potential A
A → A
= A −
∂χ
2g
(4.12)
as in Eq. 4.5 but with a meaning of χ given by Eqs. 4.8, 4.10, 4.11, whose meanings
are purely geometrical.
But because ∂χ is implied in a change of the potential, we have a hint of the fact
that the vector ω is, multiplied by a suitable physical constant (c/2 in the case of
the electron), related to the energy of the particle.
It is a first indication of the role played by the infinitesimal rotation upon itself of
the plane (n
2
, n
1
) in the geometrical interpretation of the energy associated with the
particle.
As established in Sects. 5.3 and 6.6 for the electron we will be able to insure (see
[4]) that
The infinitesimal rotation upon itself of the plane defined by the bivector spin of a
particle of spin 1/2 defines, multiplied by a suitable physical constant,(c/2inthe
case of the electron), the energy of the particle.
References
1. G.G. Jakobi, G. Lochak, C. R. Acad. Sci. (Paris) 243, 234 (1956)
2. F. Halbwachs, Théorie relativiste de fluides á spin (Gauthier-Villars, Paris, 1960)
3. D. Hestenes, J. Math. Phys. 8, 798 (1967)
4. R. Boudet, C. R. Acad. Sci. (Paris) 272A(767), (1971)
Chapter 5
Relation Between the U(1) Gauge, the Spin
and the Energy of a Particle of Spin 1/2
Abstract Real language allows one to put in evidence the relation between the U(1)
Gauge, the Spin and the Energy of a particle of spin 1/2.
Keywords Spin plane
·Infinitesimal rotation ·Momentum-energy
5.1 Relation Between the U(1) Gauge and the Bivector Spin
In the theory of the electron, the plane (n
1
, n
2
) has been called by Hestenes [1], the
“spin plane” because the bivector spin, or proper angular momentum, of the electron
is in the form “spin up”,
σ =
c
2
n
2
∧ n
1
(5.1)
In “spin down” n
2
∧ n
1
is replaced by n
1
∧ n
2
.
The relations between the U(1) gauge and the bivector spin of the electron, clearly
established in [2] and discovered independently in [1] i s absent in the standard use
of the complex formalism.
5.2 Relation Between the U(1) Gauge and the
Momentum–Energy Tensor Associated
with the Particle
The momentum–energy tensor associated with a particle in the U(1) gauge implies,
for its values v, n
1
, n
2
, a linear application from M in M in the form (see Chap. 16)
n ∈ M → N (n) = (
μ
· (i(s ∧n)))e
μ
(5.2)
R. Boudet, Quantum Mechanics in the Geometry of Space–Time,25
SpringerBriefs in Physics, DOI: 10.1007/978-3-642-19199-2_5,
© Roger Boudet 2011
26 5 Relation Between U(1) Gauge, the Spin and the Energy of a Particle
where
μ
= 2(∂
μ
R)R
−1
, s = n
3
= Re
3
R
−1
which is to be multiplied by ρg
1
, where g
1
is a suitable physical constant (equal to
c/2 in the case of the electron).
Thus N verifies (16.10)
N(v) = (∂
μ
n
2
· n
1
)e
μ
= ω (5.3)
N(n
1
) = (∂
μ
n
2
· v)e
μ
, N(n
2
) =−(∂
μ
n
1
· v)e
μ
(5.4)
which expresses the infinitesimal rotation of the sub-frame {v, n
2
, n
1
} upon itself
but in such a way that a change of gauge only affects the infinitesimal rotation of the
plane (n
2
, n
1
).
5.3 Relation Between the U(1) Gauge and the Energy
of the Particle
The energy E of the electron in a galilean frame {e
μ
} is defined by the projection on
e
0
of the vector (c/2)ω (see [3, 4])
E =
c
2
N(v) · e
0
=
c
2
ω
0
(5.5)
So one obtains a remarkable geometrical interpretation of the energy of the elec-
tron: it is the product of physical constants by the infinitesimal rotation of the “spin
plane” upon itself.
That gives a hint of the fact that in general a gauge is closely r elated to the
infinitesimal rotation of a local sub-frame upon itself.
The relation between the U(1) gauge and the interpretation of the energy is absent
in the use of the complex formalism.
References
1. D. Hestenes, J. Math. Phys. 8, 798 (1967)
2. F. Halbwachs, J.M. Souriau, J.P. Vigier, J. Phys. et le radium 22, 393 (1967)
3. R. Boudet, C.R. Ac. Sc. (Paris) 272 A, 767 (1971)
4. R. Boudet, C.R. Ac. Sc. (Paris) 278 A, 1063 (1974)
Part III
Geometrical Properties of the Dirac
Theory of the Electron