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Foundations

Parallel lines in Different Plane Geometries

End of 19th century (Frege and Peano): Efforts to axiomatize mathematics.

20th century (Russell and Whitehead): Formal theories framed in 1st Order Logic,

a method of stripping down propositions into precise formalised statements involving

only quantiﬁers (quantiﬁed variables) and their relations.

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

A Theory in 1st Order Logic

Peano Arithmetic (PA): Consists of 9 key axioms1, uses constant symbols with:

Function Symbols Relation symbols Logical symbols

+,⇥2^,_

0$,¬

S8,9,=

Example (For the Logic Enthusiasts)

Boolean expressions for De Morgan’s Laws in 1st order logic:

(¬P_Q)$(¬P)^(¬Q)

(¬P^Q)$(¬P)_(¬Q)

PA can contain arithmetic, an arithmetic statement like “2+2=4” in PA translates to:

S(S0)+S(S0)=S(S(S(S0)))

1the last axiom being the “axiom of induction”

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The ﬁrst general notion of Undecidability

PA formed basis for Prinicipia Mathematica, thought to be step towards complete

axiomatisation. But:

Theorem (1931: Gödel’s 1st Incompleteness Theorem)

Any “consistent” formal theory, strong enough to contain arithmetic (like PA ), is incomplete.

Deﬁnition (Axiomatic Independence)

A mathematical statement is undecidable in the Gödelian sense if neither it, nor its

negation, can be deduced using the rules of logic from the set of axioms being used.

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The 2nd general notion of Undecidability

Methods in Gödel’s proof (like Gödel numbering and diagonalization) vital for Church

and Turing to show undecidability of the Entscheidungsproblem in 1936.

Leads to 2nd notion of undecidability:

Deﬁnition (Decision Problem)

A problem with a YES/NO answer is undecidable if there is no algorithm that terminates

with the correct answer for the problem.

Turing Machine: Mathematical model of computation involving a

machine manipulating symbols on a bi-inﬁnite strip of tape accord-

ing to a table of rules.

Performs only one action at a time, stores no other information apart

from the state it is in and the symbols written on tape.

Universal Turing Machine (UTM): Turing machine that can simulate

any arbitrary Turing machine on any arbitrary input.

Diagram Source: http://www.cse.lehigh.edu/ glennb/um/1intro.pdf

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The Halting Problem

All of Computer Science essentially boils down to Turing’s model of computation.

Algorithms: The class of recursive functions computable by a Turing machine.

If a decision problem is intractable by a Turing Machine, it is intractable in general!

Deﬁnition (The Halting Problem)

Does a UTM halt given an initial input? Or equivalently in modern computing terms:

Given a program P and some n2N, does P eventually halt when run on input n?

Theorem (1936 Turing)

The Halting Problem is undecidable.

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Undecidability of the Halting Problem

Proof Sketch.

Consider an encoding of programs as natural numbers, and suppose there is an algorithm

for deciding when a program Phalts on an input n2N.

Write a new program Hsuch that

Hhalts on input n$program ndoes not halt on input n.

Taking n=H,His run on input Hto get contradiction:

Hhalts on input Hif and only if Hdoes not halt on input H.

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The Domino Problem

Deﬁnition (Wang Tiles)

Unit squares in the plane with coloured edges which may be translated, but not

rotated or reﬂected. Right: A Wang Tile of 2 colours.

Deﬁnition (Tileset)

A ﬁnite collection of Wang Tiles, each of which can be copied inﬁnitely

many times to tile the plane, an example on the left.

Deﬁnition (Domino Problem)

Is there a valid tiling of the entire plane using only translated copies of

the given tiles?

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Undecidability of the Domino Problem

Theorem (1966 Berger)

The Domino Problem is undecidable.

Proven by reducing a decision problem for a tileset admitting only an aperiodic tiling of

the plane to the halting problem.

Berger’s tileset of 20426 tiles reduced greatly to a set of 6 tiles, and the overall proof

simpliﬁed, by Robinson in 1966.

The Robinson Tileset

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

What is the Spectral Gap?

Deﬁnition (Hamiltonian)

A self adjoint operator on a Hilbert Space corresponding to the total p.e and k.e of a system.

Deﬁnition (Spectrum)

The set of eigenvalues of an operator A:V!V, given V is a ﬁnite dimensional vector space.

Deﬁnition (Ground State Energy and Excitation Energies)

Ground State Energy (G.S.E): Smallest eigenvalue of given Hamiltonian H⇤(L), corresponding

eigenvectors are ground states.

Excitation Energies: Eigenvalues >G.S.E, corresponding eigenvectors are excited states.

Deﬁnition (Spectral Gap)

The spectral gap (H)of Hamiltonian H⇤(L)is deﬁned as (H⇤(L)):=1(H⇤(L))0(H⇤(L)),

the difference between the smallest eigenvalues of H⇤(L).

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Spectral Gap Problem for a 2D Lattice.

Consider a translationally invariant L⇥Llattice, with each site (vertex)

assigned a complex Hilbert space H(i)'Cdof local dimension d, and

to each pair of neighbouring sites we associate a Hamiltonian h(i,j)and

a further local Hamiltonian h1(i)representing some nearest neighbour

interaction between each site. Overall, this gives us the total Hamilto-

nian:

H⇤(L):= Xh(i,j)+Xh(i)

1

H⇤(L)is gapped if (H⇤(L))(+), and gapless if spectrum above ground state is continuous.

Deﬁnition (Spectral Gap Problem)

Given a general 2 body case H⇤(L)as described as input, is H⇤(L)gapped or gapless?

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Undecidability of Spectral Gap: Proof Outline

Theorem (2015 Cubitt et al. . . )

9families of hamiltonians H⇤(L)parameterised by n 2N(essentially computable d 2⇥d2matrix

functions of parameter n) for which the Spectral Gap Problem is undecidable.

Represent structure of the Robinson tiles as classical Hamiltonian.

Add “quantum layer” on top of tiling Hamiltonian and choose suit-

able translationally-invariant coupling between layers.

Copies of a QTM encoded in a 1D history state Hamiltonian can be

effectively placed along one edge of all of the squares.

Ground state of the sandwiched Hamiltonian consists of the Robin-

son tiling conﬁguration, with computational history states in the

quantum layer.

Each of these encodes the evolution of a quantum phase estimation

algorithm and UTM.

Diagram Source: Undecidability of the Spectral

Gap by Cubbit et al, Nature 2015.

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Conclusion, Parting Thoughts and Future Directions

Analogue for Gödelian Incompleteness demonstrated in Condensed Matter Physics.

Undecidability for problems in Physics must be taken seriously, the Spectral Gap result

has been extended to even simpler 1D systems in 2018 by Bausch et al.

Yang Mills Mass Gap (a millennium prize problem) is an analogue of a Spectral Gap in

the context of QFT.

The above, along with the Haldane Conjecture may be undecidable.

Spectral Gap Undecidability has real world implications (future material applications).

Questions?

Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics