Download Solutions of the Dirac equation for spacetime

Solutions of the Dirac equation for
spacetime-dependent fields via an inverse
approach
Johannes Oertel, Ralf Schützhold
Fakultät für Physik, AG Schützhold
Universität Duisburg-Essen
DPG-Frühjahrstagung Heidelberg
March 25, 2015
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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1
Covariant Dirac equation
h
i
{γ µ , γ ν } = 2g µν ,
iγ µ ∂ µ +iqAµ − m ψ = 0,
~=c=1
Few exact analytic solutions are known, e.g.:
Constant electric field
Constant magnetic field
Constant orthogonal electric and magnetic field
Lightfront field E(x+ ),
0.8
0.6
2
Sauter pulse E(x) = E0 sech (Ωx)
1.0
0.4
√
x+ = (t + x)/ 2
0.2
0.0
-3
-2
-1
0
1
2
3
x
Depend on only one spacetime coordinate!
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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Inverse approach to the Dirac equation
Covariant Dirac equation:
h
i
iγ µ ∂ µ +iqAµ − m ψ = 0,
{γ µ , γ ν } = 2g µν
Traditional approach
Inverse approach
Fix Aµ , solve differential equation
for ψ using standard methods.
Guess ψ, solve linear equation
for Aµ .
Restrict to 1 + 1 dimensions ⇒ only 2 equations
√
Use light cone coordinates x± = (t ± x)/ 2
x−
t
x+
x
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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Dirac equation in light cone coordinates
−m√
√
2i ∂ − − 2qA−
with
A± =
√
!
!
√
2i ∂ + − 2qA+
ψ1
=0
ψ2
−m
A0 ± A1
√
,
2
∂± =
∂t ± ∂x
√
2
Two equations, easily solvable for A+ and A− , not necessarily real!
Require imaginary parts of A+ and A− to vanish
Use polar coordinates: ψk = rk eiϕk
Two equations → Eliminate r2 and ϕ2 (i.e. ψ2 )
Eliminate global phase eiϕ1 using gauge transformation
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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Result
A spinor
ψ=
±s + i
2
m
s
!
r√
∂− r
,
s=
c(x− ) −
Z
∂ − r2 dx+ −
2
(∂ r)2
m2 −
is the solution of the covariant Dirac equation with
√
m r
2 ∂+ ∂− r
qA+ = ∓ √
∓
,
m
s
2s
m s
qA− = ∓ √ ,
2r
E = ∂ − A+ − ∂ + A− .
Result
Method allows to obtain well-behaved solutions by choosing a real
function r(x+ , x− ) (ensure that s is real by adjusting c(x− )).
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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Example: Single pulses (lightfront fields)
r(x+ ) and c constant
r constant and c(x− )
E = E(x+ )
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
E = E(x− )
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Example: Two moving pulses
Combine previous two solutions: r(x+ ) and c(x− )
E=
r(x+ )
s(x− )
E− (x− ) +
E+ (x+ )
rin
sin
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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7
Example: Emerging pulses
Choose
r(x+ , x− ) = rin +
ξ
1+
e−γx+
+ e−γx−
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
.
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2 + 1 dimensions
Method can be derived in 2 + 1 dimensions as well
Solution is completely determined by real functions
r1 (x+ , x− , y), r2 (x+ , x− , y) and c(x+ , x− )
Example: Single wavefront
t
t
= −3
=0
1/m
1/m
y
1/m
2
2
1
1
y
0
1/m
-1
-2
t
=3
1/m
2
1/m
-1
-2
-2 -1 0
x
1/m
1
2
1
y
0
0
-1
-2
-2 -1 0
x
1
2
1/m
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
-2 - 1 0
x
1
2
1/m
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Conclusion
Exact solutions only for electromagnetic
fields depending on one spacetime
coordinate
Guessing solutions of the Dirac equation
leads to solutions for new pulses
Method works in 1 + 1 and 2 + 1
dimensions (possibly also in 3 + 1)
Possible usage: Starting point for
perturbation theory to investigate pair
production
Further reading: arXiv:1503.06140
[hep-th]
Slides at udue.de/oertel
Solutions of the Dirac equation for spacetime-dependent fields via an inverse approach
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10