Quantum mechanics and Noncommutative tori

Quantum mechanics as mathematics

The canonical commutation relation (CCR) in quantum mechanics (QM) is defined to the equation (with notation changed)

MP−PM= h 2πi1,

whereMis the momentum operator andP is the position operator, which are realized by unbounded self-adjoint operators on a Hilbert space as follows.

LetL²(R) be theHilbertspace of all square summable, measurable, complex- valued functions on the real line R. Let C_c,p^∞(R) be the space of all piece-wise smooth functions on R with compact support. Forf ∈L²(R), define the po- sition operator P = Mx as the multiplication operator (P f)(x) = xf(x) = Mxf(x) for x∈R. Forf ∈L²(R)∩C_c,p^∞(R), define themomentumoperator M= ₂^h_πiD as the differential operator (Mf)(x) = ₂^h_πidx^df(x) = ₂^h_πi(Df)(x) for x∈ R, which extends toL²(R) by L²-density of C_c,p^∞(R) in L²(R), where the case of one-sided derivatives at jumps inRis omitted. It then follows that for f ∈L²(R)∩C_c,p^∞(R),

(PM − MP)f(x) =x h 2πi

df

dx(x)− h 2πi

d

dx(xf(x)) =− h 2πif(x), which extends toL²(R). As well, forf, g∈L²(R), by the definitions of the L² inner product and the adjointP^∗ofP,

P^∗f, g=f, P g=

Rf(x)xg(x)dx=

Rxf(x)g(x)dx=P f, g, and henceP=P^∗ self-adjoint. Also, forf, g∈L²(R)∩C_c^∞(R),

M^∗f, g=f,Mg=

Rf(x) h 2πi

dg

dx(x)dx=

R

h 2πi

df

dx(x)g(x)dx=Mf, g, by integration by parts, and henceM=M^∗, extended toL²(R).

Moreover, we define a continuous functionf onRas, forx∈R, f(x) = _|¹_x| with |x| ≥ 1 and f(x) = 1 with |x| < 1. Then f ∈ L²(R) with L² norm 2, because

f²₂≡

R|f(x)|²dx= 2 + 2 _∞

1

1

x²dx= 2 + 2[−1

x]^∞_x=1= 4.

But

P f²₂=

R|xf(x)|²dx= 1 + 2 _∞

1

1dx= 1 + 2[x]^∞_x=1=∞. Therefore, the operatorP is not bounded, i.e., unbounded.

Furthermore, we define piece-wise smooth functionsfn onRas, for 0=n∈ Nandx∈R,fn(x) =|x|ⁿ with|x| ≤1 andfn(x) = 0 with |x|>1. Then

fn²₂= 2 ₁

0

x²ⁿdx= 2[ 1

2n+ 1x²ⁿ⁺¹]¹_x=0= 2

2n+ 1 <1.

But

Mfn²₂= 2 ₁

0

n²x²⁽ⁿ⁻¹⁾dx= 2n²[ 1

2n−1x²ⁿ⁻¹]¹x=0= 2n² 2n−1, which goes to∞, asn→ ∞. Hence, the operatorMis unbounded.

Let i = √

−1. Define the one parameter family of unitaries Ut for t ∈ R generated byM:

Ut=

Re^itλdEλ=

R

∞ k=0

(it)^k k! λ^kdEλ

= ∞ k=0

(it)^k k!

R

λ^kdEλ= ∞ k=0

1

k!(it)^kM^k = exp(itM), withU_t^∗=

Re^itλdEλ=U₋t, whereM=

Rλ dEλ

the spectral resolution for M self-adjoint and unbounded. Similarly, define Vs= exp(isP) fors∈R. It then follows by using the CCR that

VsUt= exp(isP) exp(itM) = ∞ k=0

(is)^k k! P^k

∞ l=0

(it)^l l! M^l

= ∞ k=0

∞ l=0

(is)^k k!

(it)^l l! P^kM^l

= 1 +isP+itM+(is)²

2! P²+ (is)(it)(MP− h

2πi1) + (it)² 2! M² +(is)³

3! P³+(is)²(it)

2! (MP²−2 h

2πiP) +(is)(it)²

2! (M²P−2 h 2πiM) +(it)³

3! M³+· · ·

=e^−st2hπiexp(itM) exp(isP) =e^−stiexp(itM) exp(isP) with= ₂^h_π (corrected).

If we start with M = hi_dx^d and P = Mx with [M, P] = hi1, then the exponential is given bye^sthi=e^2πist. In this case, we may setst=θ∈Rand

define theC^∗-algebra Qθ generated by those unitaries exp(isP) and exp(itM) with θ = st in the C^∗-algebra B(L²(R)) of all bounded operators on L²(R), which may be called as the quantummechanicsC^∗-algebra.

Note as well (cf. [36]) that for any Borel function f on R, the spectral integral

Rf dE for f with respect to the spectral measureE = (Et)t∈R on a Hilbert space H is defined by the strong limit

Rf dEξ= lim

n→∞

Rχ_[−n,n]f dEξ

for anyξ∈H, whereχ_[−n,n]is the characteristic function on the closed interval [−n, n] for n∈ N, and the spectral measure E = (Et)t∈R is defined by a one- parameter family of projectionsEt=E((−∞, t]) fort∈RonH, such that the following conditions are satisfied:

Operator MonotonicityEs≤Et fors < t;

Right Continuity in the strong sense Esξ= limt→s+0Etξfor anyξ∈H;

Zero 0 = limt→−∞Etξ;Identity 1 = limt→∞Etξ, for anyξ∈H.

In particular, it then follows that for a finite or infinite partition ofRwith (tj) points of partition,

R

f dE 2

=

nlim→∞

R

χ_[−n,n]f dE 2

=

⎛

⎝lim

n→∞ lim

k→∞

k>0

j=−k

χ_[−n,n](tj)f(tj)(Et_j −Et_j−1)

⎞

⎠

2

= lim

n→∞ lim

k→∞

k>0

j=−k

χ_[−n,n](tj)f(tj)²(Et_j −Et_j−1) =

R

f²dE.

In Fourier Analysis (cf. [72]), the following commutative diagram holds:

L¹(R)∩D¹(R)∩D⁻¹(L¹(R)∩D¹(R)) ^D=

d

−−−−→dt L¹(R)∩D¹(R)

∧

⏐⏐

⏐⏐^∧ C₀(R) −−−−→

M_iw C₀(R)

where thatDf ∈L¹(R)∩D¹(R) is assumed for a differentiable and summable functionf ∈L¹(R)∩D¹(R) onR, and where the Fourier transform∧is defined to be f^∧(w) =

Rf(t)e^−iwtdt∈C₀(R) the C^∗-algebra of all bounded continu- ous functions onRvanishing at infinity, forf ∈L¹(R) the Banach∗-algebra of all integrable measurable functions on Rwith convolution and involution, and D¹(R) is the algebra of all differential functions on R, and D⁻¹(·) means the inverse image by the differential operatorD. Moreover, the commutative dia- gram may be restricted toL²(R) at four corners and all the restricted corners are extended toL²(R) at four corners by taking L² closure. Furthermore, the

commutative diagram may pass to L²(S¹) by taking quotients byR (mod 1).

What’s more. the following commutative diagram holds:

L¹(R)∩D¹(R)∩M_t⁻¹(L¹(R)) −−−−→

M_t L¹(R)

⏐⏐

^{∧ ∧}⏐⏐ C₀(R)∩D¹(R) ^iD=i

d

−−−−−→dw C₀(R)

whereM_t⁻¹(·) means the inverse image by the multiplication operatorMt. More- over, the commutative diagram is restricted and extended to L²(R) at four corners, passing toL²(S¹) as well.

Example 3.5.1. (Noncommutative tori). There is the connection between quantum mechanics and the noncommutative 2-torus, as defined in the follow- ing. The noncommutative 2-torus T²θ is defined to be the universal unital C^∗-algebra generated by two unitaries U and V subject to the commutation relationV U =e^2πiθU V withe^2πiθ =λ∈Tthe 1-torus forθ∈R, so that both the spectrums ofU andV becomeT. Theuniversalityin this case means that for any unitalC^∗-algebraBgenerated by two unitaries with the same relation, there is a unital C^∗-algebra homomorphism from T²θ onto B. Note that the spectrum of a unitary operator is a compact (or closed) subset of the 1-torusT. In particular, if θ = 0 (mod 1), then T²₀ is isomorphic to the C^∗-algebra C(T²) of all continuous complex-valued functions on the 2-torus, isomorphic to theC^∗-tensor productC(T)⊗C(T) ofC(T).

Consider the unitary operators Q = Me2πix and Tθ on the Hilbert space L²(S¹) on the real 1-dimensional sphereS¹ identified withR (mod 1) homeo- morphically, defined respectively by the multiplication operator and the trans- lation operator

(Qf)(x) =e^2πixf(x) and (Tθf)(x) =f(x+θ) forf ∈L²(S¹) andx∈R(mod 1). It then follows that forf ∈L²(S¹),

(TθQf)(x) = (Qf)(x+θ) =e^2πi(x+θ)f(x+θ)

=e^2πiθe^2πix(Tθf)(x) =λ(QTθf)(x).

As well, for f, g∈L²(S¹), Q^∗f, g=f, Qg=

Rf(t)e^2πixg(x)dx=

Re^−2πixf(x)g(x)dx=Me^−2πixf, g, Tθ^∗f, g=f, Tθg=

Rf(x)g(x+θ)dx=

Rf(y−θ)g(y)dy=T₋θf, g, so that Q^∗ = Me^−2πix and T_θ^∗ = T₋θ, and moreover, Q^∗Q = QQ^∗ = 1 and T_θ^∗Tθ = TθT_θ^∗ = 1 the identity operator. Define the rotation C^∗-algebra Aθ

as the C^∗-algebra generated by these unitaries Q and Tθ in the C^∗-algebra B(L²(S¹)) of all bounded operators on L²(S¹). The rotationC^∗-algebra Aθ is

viewed as a faithful representation of the noncommutative 2-torusT²θonL²(S¹).

As well, the quantum mechanicsC^∗-algebraQθis another faithful representation ofT²θ. Namely,T²θ∼=Aθ andT²θ∼=Qθ as aC^∗-algebra. Such isomorphisms are obtained by knowing the spectrum of each unitary generator equal to the 1-torus T.

In Fourier Analysis (cf. [72]), the following dual (inner andouter) commu- tative diagram holds:

L¹(R) −−−−→^Ta

M_eiat L¹(R)

∧

⏐⏐

^{∧ ∧}⏐⏐^∧ C₀(R) −−−−−→^Ta

M_e−iaw C₀(R)

which may be restricted toL²(R) at four corners and all the restricted corners are extended toL²(R). The commutative diagram may pass toL²(S¹) by taking quotients byR(mod 1).

Denote by O(T²θ) the unital ∗-algebra C[U, V : ∅]/(V U−λU V) generated by unitaries U and V subject to the relation V U = λU V, as the coordinate ring of an algebraic noncommutative torus, as well as a dense subalgebra ofT²θ, whereC[U, V :∅] is the free algebra generated byU andV with no relation and (V U−λU V) is the two-sided ideal ofC[U, V :∅] generated byV U−λU V.

Let en(x) = e^2πinx for x ∈ R (mod 1) and n ∈ Z. Then {en}n∈Z is an orthonormal basis forL²(S¹). A positive, faithfultraceτ:T²θ→Cis defined as τ(a) =ae₀, e₀fora∈T²θ, such thatτ(ab) =τ(ba) fora, b∈T²θandτ(a^∗a)>0 ifa= 0 (cf. [60]). By using the relationsU en=en+1 andV en=e^2πiθen, check that for finite sums

m,nam,nU^mVⁿ∈ O(T²θ) witham,n∈C, τ(

m,n

am,nU^mVⁿ) =

m,n

am,nτ(U^mVⁿ)

=

m,n

am,nem, e₀=

n

a₀,n

S1

dx=

n

a₀,n. The definition for τ should be corrected as τ(a) =aδ₀, δ₀, where δ₀ means the Dirac point measure at zero (0,0), with respect to (m, n) ∈ Z², so that τ(

m,nam,nU^mVⁿ) =a₀,0, with the inner product forl²(Z²).

Sincee^2πi(θ+n)=e^2πiθ for anyn∈Z, we haveT²θ+n∼=T²θ.

Since the relationV U =λU V with λ=e^2πiθ is converted to U V =λV U, exchanging the unitary generators implies that T²θ ∼= T²₋θ ∼=T²₁₋θ. Thus may restrict the range ofθto the interval [0,¹₂]. It is known that the noncommutative toriT²θ forθ∈[0,¹₂] are mutually non-isomorphic.

•Ifθ is irrational,T²θis a simple C^∗-algebra, i.e., without no proper closed two-sided ideals, so that it has no finite dimensional representations (cf. [60]).

Proof. (Added). We use the fact later checked thatT²θis viewed as the crossed productC^∗-algebraC(T)α_θZofC(T)∼=C^∗(U) theC^∗-algebra generated by

U, by the actionαθ ofZ, defined as

αθ(U) =V U V^∗=λU =MλU =e^2πiθz=e^2πi(x+θ)∈C(T),

whereU is identified with the coordinate functionz=e^2πixonT∼=R(mod 1), so thatαθ∈Aut(T²θ) the automorphism group ofT²θ.

In general, letX be a compact Hausdorff space andGa discrete group, and letC(X)^αGbe a crossed productC^∗-algebra ofC(X) by the actionαofG.

Then any (non-trivial or not) closed two-sided idealIofC(X)αGbijectively corresponds to a non-tirivalα-invariant closed subset ofX or C(X) by taking the corresponding quotient byI.

It then easily follows that if θ is irrational, then the unique αθ-invariant closed subset ofTis justT.

•Ifθis a rational number ^p_q, with pandqrelatively prime andq >0, then T²θ has finite dimensional representations. Indeed,

Proof. (Added). Use the above proof. If θ = ^p_q, then α^q_θ = Me2πip = M₁. Therefore, there is a quotient homomorphism fromT²θ toC^qαθ Z. Moreover, the crossed productC^qαθZcontainsC^qαθZ^q as a quotientC^∗-algebra, which is isomorphic toMq(C) (cf. [3]).

What’s more (cf. [33]).

Proposition 3.5.2. Ifθ is a rational number ^p_q, with pandqrelatively prime and q > 0, then T²θ is isomorphic to the algebraC(T²,End(E))of continuous sections of the endomorphism bundle of a flat rank q complex vector bundle E on the2-torus T².

Proof. The required bundle E over T² is obtained as a quotient of the trivial bundleT²×C^q by a free action of the direct product groupG=Zq×Zq of the cyclic groupZq=Z/qZof orderq.

Consider the unitaryq×qmatrices ofMq(C) withλ=e^2πipq:

u=

⎛

⎜⎜

⎜⎜

⎝

1 0 · · · 0 0 λ . .. ... ... . .. ... ... 0 · · · · λ^q−1

⎞

⎟⎟

⎟⎟

⎠ and v=

⎛

⎜⎜

⎜⎝

0 0 · · · 1 1 0 · · · 0 ... . .. ... ...

0 · · · 1 0

⎞

⎟⎟

⎟⎠

which satisfy uu^∗ = 1q = u^∗uand vv^t = 1q =v^tv and the relations u^q = 1, v^q = 1, and

uv=

⎛

⎜⎜

⎜⎝

0 0 · · · 1 λ 0 · · · 0 ... . .. . .. ... 0 · · · λ^q−1 0

⎞

⎟⎟

⎟⎠=λ

⎛

⎜⎜

⎜⎝

0 0 · · · λ^q−1 1 0 · · · 0

... . .. . .. ... 0 · · · λ^q−2 0

⎞

⎟⎟

⎟⎠=λvu

(corrected). Define a pair (β, γ) of commuting automorphisms of orderqof the trivial vector bundleT²×C^q by sending (z₁, z₂, ξ)∈T×T×C^q to (λz₁, z₂, uξ) and (z₁, λz₂, vξ) respectively, and hence define an action ofZq×Zq. Indeed,

β^q(z₁, z₂, ξ) = (λ^qz₁, z₂, u^qξ) = (z₁, z₂, ξ), γ^q(z₁, z₂, ξ) = (z₁, λ^qz₂, v^qξ) = (z₁, z₂, ξ), γβ(z₁, z₂, ξ) = (λz₁, λz₂, vuξ)

βγ(z₁, z₂, ξ) = (λz₁, λz₂, uvξ) = (λz₁, λz₂, λvuξ)

(and hence the definition implies a pair of non-commuting automorphisms of orderq).

That action is free. Moreover, the quotient of the base space is again the torus. In this way, a flat bundleE overT² is obtained.

By definition, the space of sections of End(E) is the fixed point algebra of the induced action ofGonC(T², Mq(C))∼=C(T²)⊗Mq(C).

Using the matrix units forMq(C), we can write a section of this bundle as s=q

i,j=1fij(z₁, z₂)⊗uⁱv^j withfij(z₁, z₂)∈C(T²).

It is shown that such a section isG-invariant if and only if the coefficients ofshave the formfij(z₁^q, z₂^q).

WithinT²θwithθ= ^p_q, we haveU^qV =V U^qandV^qU =U V^q, and henceU^q and V^q belong to the center ofT²θ Any element ofT²θ has a unique expression asS =q

i,j=1fij(U^q, V^q)UⁱV^j withfij∈C(T²).

The required isomorphism is defined by sending suchSto the corresponding ssuch as above.

It follows from the proof above that the closed subalgebra generated byU^q andV^q is in fact the centerZ(T^2p

q) ofT^2p

q, so thatZ(T^2p

q)∼=C(T²).

The dense∗-subalgebraT²_θ ofT²θ forθ∈R, called as the (smooth) algebra of smooth functions on the noncommutative 2-torus, is defined by a ∈ T²_θ if a=

(m,n)∈Z²am,nU^mVⁿ, where the complex sequence (am,n) overZ²belongs to the Schwartz spaceS(Z²) of rapidly decreasing sequences overZ², such that

sup

m,n∈Z(1 +m²+n²)^k|am,n|<∞, k∈N.

Note that in the case ofθ= 0, forf ∈C(T²), the inverse Fourier transformf^∨ off belongs toS(Z²) if and only iff belong toC^∞(T²) the algebra of smooth functions onT². May prove it, but no time at this moment.

Note that the (classical)Fourier transform∧:L¹(Z²)→C(T²) is defined as

g^∧(z₁, z₂) =

m,n∈Z

g(m, n)z₁^mz₂ⁿ

forg∈L¹(Z²) the Banach ∗-algebra of integrable functions on Z²with convo- lution and involution. The Fourier transform extends to theC^∗-algebra isomor- phism from the groupC^∗-algebraC^∗(Z²) toC(T²). As well, the inverse Fourier

transform is defined as f^∨(m, n) =

T²

f(z₁, z₂)z₁^mz₂ⁿdz₁dz₂, f ∈C(T²).

May prove that the functionf^∨defined so belongs toC^∗(Z²).

There are the following inclusions with settings as

O(T²θ) =P(T²)θ⊂T²_θ=C^∞(T²)θ⊂T²θ=C(T²)θ

resembling those of algebras of algebraic functions or polynomials of coordinates, of smooth functions, and of continuous functions onT², at θ= 0.

It is shown that if θ = ^p_q a rational, then T²p

q is isomorphic to the space C^∞(T²,End(E)) of smooth sections of the bundle End(E) overT².

A derivation on a complex algebraA is defined to be a C-linear map δ : A→Asuch that δ(ab) =δ(a)b+aδ(b) for all a, b∈A as the product formula of differentiation of functions. A derivation on A is determined by values of generators ofAunderδ(and in fact, as well as those of their products assigned by computation, necessary to define the derivation).

A∗-derivation on an involutive algebraAoverCis a derivationδonAsuch thatδ(a^∗) =δ(a)^∗ for anya∈A.

In particular, if δ is a derivation on a unital algebra with the unit 1, then δ(1) =δ(1)1 + 1δ(1) = 2δ(1). Therefore,δ(1) = 0.

Example 3.5.3. First define (involutive) linear maps δ₁, δ₂ : T²_θ → T²_θ as ∗- derivations byδ₁(U) = 2πiU andδ₁(V) = 0 andδ₂(U) = 0 andδ₂(V) = 2πiV. Next define as that

δ₁(U²) =δ₁(U)U+U δ₁(U) = 2(2πi)U², δ₁(V²) = 0, δ₁(U V) =δ₁(U)V +U δ₁(V) = (2πiU)V = 2πiU V, δ₁(V U) =δ₁(λU V) =λ2πiU V = 2πiV U,

δ₂(U²) = 0, δ₂(V²) = 2(2πi)V²,

δ₂(U V) =δ₂(U)V +U δ₂(V) =U(2πiV) = 2πiU V, δ₂(V U) = 2πiV U.

Also define as that

δ₁(U³) =δ₁(U²)U+U²δ₁(U) = 3(2πi)U³, δ₁(V³) = 0, δ₁(U²V) =δ₁(U²)V +U δ₁(V) = 2(2πi)U²V,

δ₁(V U²) =V δ₁(U²) = 2(2πi)V U²,

δ₁(U V²) =δ₁(U)V²+U δ₁(V²) = 2πiU V², δ₁(V²U) =V²δ₁(U) = 2πiV²U,

δ₂(U³) = 0, δ₂(V³) = 3(2πi)V³, δ₂(U²V) =δ₂(U²)V +U²δ₂(V) = 2πiU²V, δ₂(U V²) =δ₂(U)V²+U δ₂(V²) = 2(2πi)U V², δ₂(V U²) = 2πiV U², δ₂(V²U) = 2(2πi)V²U.

Inductively, assigning the values of the productsU^mVⁿ for m, n∈Zunder δ₁ andδ₂, by thatU^∗=U⁻¹,V^∗=V⁻¹, andδj(a^∗) =δj(a)^∗fora∈T²_θ,j= 1,2, we obtain that with am,n ∈ C for m, n ∈ Z, but finitely many, or not, with (am,n)∈S(Z²),

δ₁

⎛

⎝

m,n∈Z

am,nU^mVⁿ

⎞

⎠ = 2πi

m>0,n∈Z

mam,nU^mVⁿ−2πi

m<0,n∈Z

mam,nU^mVⁿ,

δ₂

⎛

⎝

m,n∈Z

am,nU^mVⁿ

⎞

⎠ = 2πi

m∈Z,n>0

nam,nU^mVⁿ−2πi

m∈Z,n<0

nam,nU^mVⁿ (corrected).

The traceτ onT²θ defined asτ(

m,nam,n) =a₀,0has the invariance prop- erty as a noncommutative analogue of the invariance property of the Haar mea- sure for the torus, as thatτ(δj(a)) = 0 for anya∈T²_θandj= 1,2.

Indeed, it is shown by computation above thatδj(a) has the zero component a₀,0 = 0 at (0,0) because of killing the constants as δj(1) = 0 as the usual differentiation for functions.

Those ∗-derivations δj on T²_θ generate commuting one-parameter group of

∗-automorphismsαj(t) ofT²θ fort∈R.

Namely, defineαj(t) = exp(tδj) onT²_θ. Check that δ₂(δ₁(

m,n∈Z

am,nU^mVⁿ))

=−4π²

m>0,n>0

mnam,nU^mVⁿ+ 4π²

m>0,n<0

mnam,nU^mVⁿ

+ 4π²

m<0,n>0

mnam,nU^mVⁿ−4π²

m<0,n<0

mnam,nU^mVⁿ,

=δ₁(δ₂(

m,n∈Z

am,nU^mVⁿ)),

and hence δ₁ commutes with δ₂ onT²_θ, so that α₁(t) commutes with α₂(t) on T²_θ, and moreover, since αj(t) are unitaries, they extends to those on T²θ by continuity.

As well, a continuous actionαof the 2-torusT²onT²θis defined asαz₁,z₂U = z₁U andαz₁,z₂V =z₂V for (z₁, z₂)∈T².

Note that by definition, exp(tδ₁)(U) =e^2πitU,α₁(t)V = 0, andα₂(t)U = 0, exp(tδ₂)V =e^2πitV.

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