Applications of
PT-SYMMETRIC QUANTUM
MECHANICS
Carl Bender
Washington University
Lecture #2
THINGS TO BE DISCUSSED:
I. Analytic continuation of eigenvalue problems II. Construction of the secret symmetry operator C and the Hilbert-space metric
III. Strange features of bound states
IV. Fast time evolution and the quantum brachistochrone V. Ghost states in quantum field theory
VI. Double-scaling limit
Topic I. Analytic continuation of
Eigenvalue problems
Two puzzles…
The anharmonic oscillator and all that
How general is the PT phase transition?
Implicitly restarted Arnoldi algorithm
CMB and D. Weir
[arXiv: quant-ph/1206.5100] Journal of Physics A (in press)
Phase transition at g = 0.04
Topic II. The secret symmetry operator.
The eigenvalues are real ...
But is this quantum mechanics??
• Probabilistic interpretation??
• Hilbert space with a positive metric??
• Unitarity??
In ordinary Hermitian quantum mechanics:
One might think that in PT quantum
mechanics the inner product is:
But now the metric is not positive:
P. A. M. Dirac: Bakerian Lecture,
Proceedings of the Royal Society A (1941)
How to construct the
Hilbert- space inner product…
The Hamiltonian determines its own adjoint!
Unitarity
With respect to the CPT adjoint
the theory has UNITARY time
evolution.
Norms are strictly positive!
Probability is conserved!
Example: 2 x 2 Non-Hermitian
matrix PT -symmetric Hamiltonian
where
Topic III. Strange features of bound
states
Topic IV. How do we interpret a
non-Hermitian Hamiltonian??
Answer: Solve the quantum brachistochrone problem…
Classical brachistochrone
• Newton
• Bernoulli
• Leibniz
• L'Hôpital
Classical brachistochrone
is a cycloid
Gravitational field
Quantum brachistochrone
Constraint:
Hermitian case
becomes
Minimize t over all positive r
while maintaining constraint
Minimum evolution time:
Looks like uncertainty principle but is merely rate times time = distance
Non-Hermitian PT -symmetric
Hamiltonian
where
Exponentiate H
The bottom line…
What does PT symmetry really mean?
Interpretation…
Finding the optimal PT-symmetric
Hamiltonian amounts to constructing
a wormhole in Hilbert space!
“The shortest path between two
truths in the real domain passes
through the complex domain.”
-- Jacques Hadamard
The Mathematical
Intelligencer 13 (1991)
Quantum State Discrimination…
a closely related topic
Topic V. Field theoretic
applications...ghosts!
Lee Model
The problem with the Lee Model:
“A non-Hermitian Hamiltonian is unacceptable
partly because it may lead to complex energy
eigenvalues, but chiefly because it implies a non-
unitary S matrix, which fails to conserve probability
and makes a hash of the physical interpretation.”
PT quantum mechanics to the rescue…
Meep! Meep!
PT
GHOSTBUSTING:
Reviving quantum
theories that were thought
to be dead
Gives a fourth-order field equation:
Pais-Uhlenbeck action
The problem: A fourth- order field
equation gives a propagator like
GHOST!
Two possible realizations…
There can be other realizations as well!
Calculate the equivalent Dirac
Hermitian Hamiltonian:
No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck
model, CMB and P. Mannheim, Physical Review Letters 100, 110402 (2008) CMB and P. Mannheim, Physical Review D 78, 025002 (2008)
How does PT symmetry work in
QFT? Feynman Rules for g f 3
Vertex
Line
Vacuum graphs of order g 2
D D
Vacuum graphs of order g 4
D D
+ many more!
Perturbation series for free energy
(ground-state energy) F :
Divergent and NOT Borel summable ---
all coefficients in series have same sign.
There is a discontinuity across cut in g plane ---
so vacuum energy F is complex.
This means that vacuum state is unstable.
Feynman Rules for ig f theory 3
Vertex
Line
Now, series for F alternates in sign, and
is Borel summable.
Free energy is real , vacuum state is stable!
Path of functional integration
(convergent!) Note: left-right (PT) symmetry
Perturbative saddle point
Feynman Rules for g f theory 4
Vertex
Line
Vacuum graph of order g
Vacuum graphs of order g 2
Perturbation series for free energy F :
Alternates in sign and is Borel summable.
Free energy is real and ground state is stable.
Feynman Rules for - g f theory 4
Vertex
Line
Oops! Now, the series for the free energy
does NOT alternate in sign. Is the ground
state unstable? NO, IT IS STABLE!
Perturbative and
nonperturbative saddle points
Path of functional
Integration (convergent!)
Note: left-right (PT) symmetry
Topic VI. Resolution of ambiguity in
Double-scaling limit
PT quantum mechanics is fun!
You can re-visit things you
already know about ordinary
Hermitian quantum mechanics.
Possible fundamental applications:
1. PT Higgs model: theory is asymptotically
free, stable, conformally invariant, and has
2. PT QED like a theory of magnetic charge,
asymptotically free, opposite Coulomb force
3. PT gravity has a repulsive force
4. PT Dirac equation allows for massless neutrinos
to undergo oscillations
The end !
I hope you enjoyed the lectures.
