2. Classical realism
“If, without in any way disturbing a system,
we can predict with certainty ( ) the value of ...
a physical quantity, then there exists an
element of physical reality corresponding to
this physical quantity.”
A. Einstein, B. Podolsky, and N. Rosen Phys. Rev 47,777 (1935).
Element of physical reality → Observable → Real Number
2
3. Normed division algebras
● Only 4 such mathematical objects:
– Real Numbers: R
– Complex Numbers C (maximum numbers of
properties)
– Quaternions H (non-commutative)
– Octonions O(non-commutative and
non-associative)
● Q1: Why not R?
● Q2: If not R, why C?
3
4. C is “strange”
● What does sqrt(-1) mean ?
● Matrix representation:
a + ib = ( a b)
(-b a)
i=(01) 1=(10) i*i = (-1 0)
(-1 0 ) (01) ( 0 -1)
4
5. Textbook Quantum Mechanics
●
State vector Ψ in a Hilbert space.
●
Schrödinger equation:
●
|Ψ| is a probability amplitude.
2
●
No more classical local realism.
5
6. De Broglie-Bohm's Approach
●
Decompose Ψ in real and imaginary parts.
●
Particle has definite position and mo-
mentum.
●
“Pilot wave” - derived from Ψ.
●
Non-relativistic effects (instantaneous response).
●
Inspired Bell to discover his inequalities.
●
Saves realism at the expense of non-locality.
– NOT THE WAY TO GO 6
7. Proposed Strategy
● Look for the common features of classical and
quantum mechanics.
– Bohr: “The correspondence principle has more secrets to reveal”.
– Aage Petersen and Emile Grgin: Quantal Algebra.
● Will generate 2 intuitive principles from which
only classical and quantum mechanics will follow.
● A third separation principle is needed. Any
principle will do, choose the simplest one.
7
8. Axioms of Quantum Mechanics
1. The Composability Principle: All physical systems
can interact with each other.
– There are no islands isolated universes.
2. The Consistency Principle: One can reason
consistently about the physical system.
3. The Separation Axiom: quantum mechanics is not
classical mechanics.
– Principles 1 and 2 imply classical and quantum mechanics, and no other solution
is possible.
– Principles 1 and 2 imply both non-relativistic and relativistic physics.
– Any pure quantum effects can be postulated here. This is the most intuitive
separation axiom.
– Future research: prove axiom 3 is not needed, and only quantum mechanics can 8
construct a consistent theory of nature. Hint: Classical general relativity runs
into problems without quantum mechanics.
9. Quantal Algebra
What are the characteristics of both classical and
quantum mechanics?
– They have an anti-symmetric product:
● Poisson bracket (CM).
● Commutator (QM).
– They have a symmetric product:
● Regular function multiplication (CM).
● Jordan product (anti-commutator) (QM).
– Composability implies preservation of the 2
products under a tensor product.
● The syntax of how we describe nature is the same for individual
systems and for the composed system. In other words, if classical
mechanics describes the Earth and the Sun, the composed system
is described under the same formalism).
● Composability has trivial consequences for classical mechanics: the 9
additivity of the phase space; but there is a non-trivial solution in
the case of quantum mechanics.
10. Preliminaries: states, observables,
and generators
●
Ψ defines a state.
– A state describes the observer's knowledge. It is a relational object, like speed.
– Wavefunction collapse reflects the change in the observer's knowledge.
– States may be pure or mixed.
– States live in a Hilbert space.
●
Observables are Hermitian operators acting on states.
– Observables form an algebra called a Jordan algebra.
– A Jordan algebra defines elements of reality.
●
Generators are anti-Hermitian operators acting on
States.
– Generators form a Lie algebra.
10
– A Lie algebra defines time evolution.
11. Preliminaries: states and Jordan
algebras
● All states (pure and mixed) form a convex set.
● The pure states form the boundary of the convex set.
● The pure states are related to projection operators: P*P=P and questions one
may ask a physical system. (ask the same question twice in a row and the
answer does not change).
● Piron's result: from elementary questions recover the geometry of the Hilbert
space (over the reals, complex, and quaternionic numbers).
● Observables corresponding to pure states form an algebra with 2 properties:
– XY = YX
– (XY)(XX) = X(Y(XX)) (Jordan identity)
● Jordan algebra is not associative, but power associative (a weaker property)
– Observables in QM are not associative
– Observables in CM are associative
● Three important infinite families of Jordan algebras:
– The Jordan algebra of nXn self-adjoined real matrices.
– The Jordan algebra of nXn self-adjoined complex matrices. 11
– The Jordan algebra of nXn self-adjoined quaternionic matrices.
12. Preliminaries: Lie groups and Lie
algebras
●
Symmetries are mathematically described as groups.
●
A symmetry can be discrete or continuous.
●
A continuous group is called a Lie group (after Sophus Lie
- a 19th century Norwegian mathematician).
●
A Lie group is also a manifold. The tangent space at the
identity element defines a linear space called a Lie algebra.
●
The most important mathematical object in Lie algebras is
the commutator. The commutator fully characterizes the
algebra.
●
The commutator obeys the Jacobi identity:
[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0
12
13. How to obtain Quantum
Mechanics from the three
postulates?
●
Two methods: indirect and direct ones.
●
Indirect Method:
– Based on Lie group dimensional analysis.
– Three classical infinite families of Lie
groups:
● Orthogonal groups O(n): n(n-1)/2
● Symplectic groups Sp(n): n(n+1)/2
● Unitary groups: U(n): n*n 13
14. Indirect method
● How many distinct elementary questions N one may
ask of a physical system with n states?
● Use composability. Prove that N = n^r with r=1,2,3,...
● r = 1 corresponds to classical mechanics.
● r = 2 corresponds to non relativistic quantum mechanics.
N= n*n corresponds to U(n) unitary group. (there are n*n
projector operators constructible from |Ψ1, Ψ2, ..., Ψn>).
● r>2 implies sudden jumps in probabilities. In fact r=3
corresponds to second quantization and field theory and
there is nothing beyond 3.
– Think of the Feynman diagram and particle
creation and annihilation. This corresponds to
sudden jumps in probabilities.
14
15. Consistent reasoning about nature
●
R = 1: classical mechanics and Kolmogorov statistical
theory. (Also obeys all Bell's inequalities).
●
R = 2: Born rule: r is the valence of the inner product:
<Ψ|Ψ> = a scalar (count 1 for a vector and 0 for a
scalar. Total = 2 = r).
– This leads to a Hilbert space.
●
R = 3: Zovko rule: r is the valence of the inner
product: <Q|Q> = a future oriented Minkowski
vector. (count 1 for a vector and 0 for a scalar. Total =
3 = r).
– This leads to a Connes space (non-
15
commutative geometry).
16. Direct method
●
Let A be system 1, B be system 2.
●
Let Sa be the symmetric product of system A, and Sb be the
symmetric product of system B.
●
Let Aa and Ab be the antisymmetric products.
●
Let St and At be the symmetric and antisymmetric products of the
total composite system.
●
Composability implies:
– At = Sa Ab + Sb Aa
– St = Sa Sb + λ Aa Ab
With λ = -1, 0, 1.
●
λ = 0 corresponds to classical mechanics. (parabolic case).
●
λ = -1 corresponds to quantum mechanics. (elliptic case).
●
λ = +1 corresponds to quantum mechanics based on 16
split-complex numbers (J*J = +1). (hyperbolic case).
17. Direct method (2)
●
λ = +1 does not satisfy von Neumann uniqueness theorem and hence
does not satisfy postulate 2 and is eliminated.
●
λ = -1 corresponds to quantum mechanics based on complex
numbers. THIS IS THE ORIGIN OF COMPLEX NUMBERS IN QUANTUM
MECHANICS.
●
Compare:
– At = Sa Ab + Sb Aa
– St = Sa Sb - Aa Ab
with
a + ib = ( a b)
(-b a)
and
(Sa Aa) * (Sb Ab) = (Sa Sb - Aa Ab Sa Ab + Aa Sb) 17
(-Aa Sa) (-Ab Sb) (-Aa Sb – Sa Ab -Aa Ab + Sa Sb)
18. Axioms revisited
1. The Composability Principle: All physical systems
can interact with each other.
2. The Consistency Principle: One can reason
consistently about the physical system.
3. The Separation Axiom: quantum mechanics is not
classical mechanics.
Composability gives λ = -1, 0, 1.
Consistency eliminates λ = +1.
Separation eliminates λ = 0.
λ = -1 coresponds to both non-relativistic (r = 2) and relativistic 18
(r=3) quantum mechanics. (λ = -ħ*ħ/4)
19. Active Research Project
●
Prove Axiom 3 is not needed.
●
One may construct a self-consistent description of nature
only in quantum mechanical terms.
– Hints: general relativity (a pure classical theory) is
flawed: contains singularities and contains many
time travel solutions.
●
Other forms of Axiom 3:
– “the maximum extractable information from a
physical system is bounded from above”.
– “state space is continuous”.
“quantum mechanics is inherently random”.
19
–
20. Quantal Algebra
● Combines Lie and Jordan algebra in the same
structure. Call symmetric product σ,
antisymmetric product α.
● Three identities:
(fαg)αh + (gαh)αf + (hαf)αg = 0 (Jacobi)
gα(fσh) = (gαf)σh + fσ(gαh) (Leibniz)
(fσg)σh − fσ(gσh) = -λgα(hαf) (Petersen)
Leibnitz: Suppose T = I + εFα is an infinitesimal
automorphysm. σ product must be compatible with
α: 20
T (fσg) = (T f)σ(T g).
21. C* algebra
● Define an associative product * out of σ and α:
– F*G=FσG+iFαG
● * acts on states (C* or von Neumann algebra).
● σ acts on observables (Jordan-Banach or Jordan-
Banach-Wigner algebra).
● α acts on generators (Lie algebras).
● C* algebra cannot be a R* of H* algebra:
– Quaternions H do not satisfy the Leibniz identity;
– Reals are related to second quantization and bosons;
– Quaternions are related to second quantization and fermions;
– Bosons and fermions do not satisfy composability: 2 fermions are not 21
another fermion.
22. Conclusions
● Quantum mechanics mixes observables with
generators.
● This mixing is at the root of QM peculiarity:
– Violates Bell's inequalities. (Bohm's
interpretation does not violate the
inequalities and is incompatible with QM).
– Demands complex numbers.
● Complex numbers and observables-generators
mixing correspond in state space to Kahler manifolds
and maximum entanglement.
● No generalization of QM is possible.
● Non-relativistic QM is fully understood, relativistic QM 22
is work in progress.
