Eigen Equation of the Nonlinear Spinor
arXiv:0704.0436v2  [physics.gen-ph]  4 Apr 2007
Eigen Equation of the Nonlinear Spinor
Ying-Qiu Gu∗ Ta-tsien Li†
Department of Mathematics, Fudan University, Shanghai, 200433, China
Abstract
How to effectively solve the eigen solutions of the nonlinear spinor field equa-
tion coupling with some other interaction fields is important to understand the
behavior of the elementary particles. In this paper, we derive a simplified form
of the eigen equation of the nonlinear spinor, and then propose a scheme to
solve their numerical solutions. This simplified equation has elegant and neat
structure, which is more convenient for both theoretical analysis and numerical
computation.
PACS numbers: 11.10.Ef, 11.10.Lm, 11.10.-z
Key Words: nonlinear Dirac equation, nonlinear spinor field,
eigen equation, quaternion
1 Introduction
Almost all the elementary fermions have spin-1
, which can be naturally described by
spinors, so today spinors and spinor representations play a more and more important
role in mathematical and theoretical physics. Noticing the limitations of the linear field
equation, many physicists such as H. Weyl, W. Heisenberg, once proposed the nonlinear
spinor equations[1, 2, 3, 4, 5, 6] to construct a unified field theory for elementary
particles. However they have not gotten many definite results due to the mathematical
difficulties. The rigorous solutions for some simple dark nonlinear spinor models were
∗email: yqgu@luody.com.cn
†email: dqli@fudan.edu.cn
http://arxiv.org/abs/0704.0436v2
obtained in [7, 8, 9, 10], and we found it can provide negative pressure to guarantee a
singularity-free and accelerating expanding universe[11, 12].
The theoretical proof about the existence of solitons was investigated in [13, 14,
15, 16, 17, 18]. The symmetries and many beautiful conditional exact solutions of the
nonlinear spinor, vector and scalar differential equations are collected in [19]. However
lots of these exact solutions seem to be non-physical.
The spinor with its own electromagnetic potential was studied in [20, 21, 22, 23, 24],
it was disclosed that the nonlinear spinor equations have particle like solution with
anomalous magneton, and imply the exact classical mechanics and quantum mechanics
for many-body[25, 26, 27].
In this paper we derive a simplified form of the eigen equation with general meaning
for nonlinear Dirac equation, and then give a scheme to solve the solution. Denote the
Minkowski metric by ηµν = diag[1,−1,−1,−1], Pauli matrices by
~σ = (σj) =
. (1.1)
Define 4× 4 Hermitian matrices as follows
, γ =
, β =
0 −iI
. (1.2)
In this paper, we adopt the Hermitian matrices (1.2) instead of Dirac matrices γµ,
because this form is more convenient for calculation.
For the system of a nonlinear spinor field φ in the 4−d potential Aµ, the Lagrangian
describing the motion is generally given by
L = φ+[αµ(~i∂µ − eAµ)− µcγ]φ+ F (γ̌, β̌), (1.3)
where µ > 0 is a constant mass, F are the nonlinear coupling potential, which is usually
the even polynomial of γ̌, β̌, and γ̌, β̌ are the quadratic scalars of φ defined by
γ̌ = φ+γφ, β̌ = φ+βφ. (1.4)
We can check γ̌ is a true-scalar, but β̌ a pseudo-scalar.
The variation of (1.3) with respect to φ+ gives the dynamic equation
αµ(~i∂µ − eAµ)φ = (µcγ − Fγγ − Fββ)φ, (1.5)
where Fγ =
, Fβ =
. In the Hamiltonian form we have
~i∂tφ = Ĥφ, Ĥ ≡ ~α · (−~i∇− e ~A) + eA0 + (µc− Fγ)γ − Fββ. (1.6)
In this paper we denote ~A = (A1, A2, A3) to be the spatial part of a contravariant
vector Aµ.
It is easy to check that the current conservation law holds ∂µq
µ = 0, so we can take
the normalizing condition as follows
|φ|2d3x = 1. (1.7)
The 4− d potential produced by spinor φ itself takes the following form
µ = eα̌µ = eφ+αµφ. (1.8)
2 Simplification of the Equation
Consider a spinor keeping motionless in an external magnetic field ~B = (0, 0, B). since
the scale of the elementary particle is very small, we take external field B as a constant.
By ∇× ~Aext = ~B, we have the general solution for external vector potential
~Aext =
B(−y, x, 0) +∇Φ, (2.1)
where Φ(~x) is any given smooth function.
In the spherical coordinate system (r, θ, ϕ), we have
~σ · ∇ = σr∂r + (σθ∂θ + σϕ∂ϕ), (2.2)
where (σr, σθ, σϕ) is given by
cos θ sin θe−ϕi
sin θeϕi − cos θ
− sin θ cos θe−ϕi
cos θeϕi sin θ
r sin θ
0 −ie−ϕi
ieϕi 0
. (2.3)
Let Ĵ be the angular momentum operator for the spinor field
Ĵ = ~r × (−~i∇) +
~~S, ~S = diag(~σ, ~σ), (2.4)
then any eigenfunction of Ĵ3 = −~i∂ϕ +
~S3 takes the following form
φ = (u1, u2e
,−iv1,−iv2e
ϕi)T exp
(2.5)
with (κ = 0,±1,±2, · · · ), where uk, vk(k = 1, 2) are functions of r, θ but independent
on ϕ and t. In this paper the index T stands for transposed matrix.
For any spin-1
particle, it has a pole axis. If we set the pole axis as coordinate
x3 = z, then Ĵ3 is commutative with the nonlinear Hamilton operator (1.6) by a U(1)
gauge transformation for spinor as eΦiφ, which removes the uncertain function from
external vector potential (2.1), thereby we have
~Aext =
B(−y, x, 0) =
Br sin θ(− sinϕ, cosϕ, 0). (2.6)
For the above symmetric form of vector potential (2.6), substituting (2.5) into (1.6) we
can check that all functions uk, vk can take real number. For the vector potential pro-
duced by φ itself, we will find below it also takes the form of (2.6). This simplification
of φ may be the essence of the gauge symmetry.
In what follows, we set ~ = c = 1 as units for convenience. Making variable
transformation
u = u1(r, θ) + u2(r, θ)i, v = v1(r, θ) + v2(r, θ)i, (2.7)
then we have {
α̌0 = |u|
2 + |v|2, α̌ = (ūv − uv̄)i,
γ̌ = |u|2 − |v|2, β̌ = ūv + uv̄,
(2.8)
with ~̌α = α̌(− sinϕ, cosϕ, 0). Substituting (2.7) and (2.8) into (1.3) we get the La-
grangian of the eigen states as follows
L = Re
ū(∂r +
∂θ)v̄ − v̄(∂r +
∂θ)ū
r sin θ
(κ+ 1
)(ūv − uv̄)
+(m− eA0)(|u|
2 + |v|2)− ie(ūv − uv̄)A− µ(|u|2 − |v|2) + F,
(2.9)
where
(~r × ~A) · ez = (cosϕAy − sinϕAx) = A(r, sin θ) (2.10)
including both external and inner vector potential. By variation with respect to ū, v̄,
we get an elegant equation with double-helix structure
eθi(∂r +
∂θ)ū =
r sin θ
[(κ+ 1
)u− 1
ū] + (µ+m− eA0 − Fγ)v + Fβu+ ieAu,
eθi(∂r +
∂θ)v̄ =
r sin θ
[(κ + 1
)v − 1
v̄ ] + (µ−m+ eA0 − Fγ)u− Fβv + ieAv.
(2.11)
By (2.11) we find that, κ = 0 corresponds to spin 1
and κ = −1 corresponds to spin
. Generally the coordinates r and θ can not be separable for nonlinear equation.
The energy functional for (2.11) is given by
E = 2π
r2 sin θdrdθ
ū(∂r +
∂θ)v̄ − v̄(∂r +
∂θ)ū
r sin θ
(κ+ 1
) + eA
i(ūv − uv̄) + eA0(|u|
2 + |v|2) + µ(|u|2 − |v|2)
(2.12)
The eigen solution of (2.11) is just the extreme point of E under the constraint of
normalizing condition
(|u|2 + |v|2)r2 sin θdrdθ = 1. (2.13)
(2.13) is also the quantizing condition of the energy spectrum[9, 10].
3 A Scheme for Solving Solution
For general potential Aµ and F , the analytic solution of (2.11) u and v can not be
solved. However they can be conveniently expressed by Fourier series of θ, and the
equations of the radial functions can be derived by variation principle. For any given
integer N ≥ 0, define 2N + 1 vectors
Γ(θ) = (e−2Nθi, e−2(N−1)θi, · · · , e2(N−1)θi, e2Nθi), (3.1)
U(r) = (U−N(r), U−(N−1)(r), · · · , U(N−1)(r), UN(r))
, (3.2)
V (r) = (V−N(r), V−(N−1)(r), · · · , V(N−1)(r), VN(r))
T . (3.3)
The eigen solution of (2.11) with even parity must take the form
u = Γ · U =
2nθi, v = Γ̄ · V eθi =
(−2n+1)θi, (3.4)
and the eigen solution with odd parity takes
u = Γ · Ueθi =
(2n+1)θi, v = Γ̄ · V =
−2nθi. (3.5)
In what follows we only consider (3.4). For (3.5) we have similar results.
For the cases κ 6= 0 and κ 6= −1, i.e. for the cases with nonzero magnetic quantum
number, the solution must have consistent conditions at θ = 0, π as
u(r, 0) = u(r, π) =
Un(r) ≡ 0, v(r, 0) = v(r, π) =
Vn(r) ≡ 0. (3.6)
For this case, (3.4) minus (3.6) we get
2nθi − 1), v =
−2nθi − 1)eθi. (3.7)
By the form (3.7) we have
e2nθi − 1
2i sin θ
= (1 + e2θi + e4θi + · · ·+ e2(n−1)θi)eθi, (3.8)
which removes the singularity of (2.11) at θ = 0, π.
For the spin 1
state, i.e. for the case κ = 0, we can check from (2.11) that the
solution U, V are all real functions. But for the spin −1
state, i.e. for the case κ = −1,
the solution U, V are all pure imaginary functions. In what follows we only consider
the real case. For the covariant quadratic forms (2.8), by (3.3) we have
α̌0 = U
TPU + V T P̄ V, α̌ = UT (Q+ −Q)V i,
γ̌ = UTPU − V T P̄V, β̌ = UT (Q+ +Q)V,
(3.9)
where P = Γ+Γ and Q = ΓTΓeθi are (2N +1)× (2N +1) matrices with components as
Pm,n = exp[2(n−m)θi], (−N ≤ n,m ≤ N),
Qm,n = exp[(2(n+m) + 1)θi].
(3.10)
By (3.9) and (3.10) we have
α̌0 =
n,m=−N
(UnUm + VnVm)(e
−2(n−m)θi + e2(n−m)θi), (3.11)
n,m=−N
(UnUm − VnVm)(e
−2(n−m)θi + e2(n−m)θi), (3.12)
n,m=−N
UnVm(e
−2(n−m)θi − e2(n−m−1)θi)eθii, (3.13)
n,m=−N
UnVm(e
−2(n−m)θi + e2(n−m−1)θi)eθi. (3.14)
The dynamic equation of the potential A is given by
−∆A = eα̌ = e[a0(r) sin θ + a1(r) sin 3θ + · · · ],
= e[a0(r)(e
−2θi − 1) + a1(r)(e
−4θi − e2θi) + · · · ]eθii. (3.15)
Substituting (3.4), (3.7), (3.8) and (3.11)∼(3.15) into (2.11) and directly comparing
the coefficients of all e2nθi, we can easily get a truncated ordinary differential equation
of U(r), V (r). However the convergence this cut-off equation to the original solution
needs proof. A more credible method to get the efficient equation of U(r), V (r) is
via variational principle. The variational equation can be obtained by the following
procedure. Define operators
T̂u =
ΓT (θ)e−θi sin θdθ, T̂v =
Γ+(θ) sin θdθ, (3.16)
then T̂u left multiplying the first equation of (2.11) and T̂v left multiplying the second
give the variational equation of U(r), V (r). The coefficient matrix of U ′(r) and V ′(r)
is the same positive definite symmetric matrix with components
Mn,m =
Pn,m sin θdθ =
1− 4(n−m)2
. (3.17)
The other coefficient matrices can also be similarly obtained.
The convergence of expansion (3.4) and the consistent condition (3.6) seem to
have closely relation with the structure of nonlinear potential F (γ̌, β̌), which reflects
the important properties of the elementary particles such as the exclusion princi-
ple. The above procedure is valid for extensive models. The neat and elegant re-
sults are profoundly rooted in the quaternionic structure of the physical variables and
spacetime[28, 29, 30], so the 3 + 1 dimensional Universe is a miraculous masterpiece
with unique feature.
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