Quantum physics and the time operator - eonet.ne.jptfujimoto21/Desy2.pdf · Quantum physics and the time operator Tadashi FUJIMOTO, Dept.Philosophy Ryukoku Univ. Kyoto AQFT Seminar:

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Quantum physics and the time operator

Tadashi FUJIMOTO, Dept.Philosophy

Ryukoku Univ. Kyoto

AQFT Seminar: 07.12.2011 at Hamburg University

Tadashi FUJIMOTO, Dept.Philosophy (ryuukoku)Quantum physics and the time operatorAQFT Seminar: 07.12.2011 at Hamburg University 1

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Table of Contents

Contents

.. .1 Motivation

.. .2 Three types of time in quantum physics

.. .3 History of the time operator

.. .4 Classification of time operators according to CCR

.. .5 Examples of time operators according to the above classification

.. .6 Positive operator valued measure (POVM)

.. .7 Summary and future plans

.. .8 Philosophical essay


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Motivation

Motivation

Philosophical interestIn our world, we can recognize “time” through spatial movement(ex . watch, clock ,movement of the sun or moon). But fromwhere comes this time? (cf, St.Augustinus (4th cent.), I.Kant,H.Bergson, M.Heidegger, etc. )

Mathematical interest and physical interestFrom the viewpoint of operator theory, what characteristics doesthe time operator have and what kind of time operators arethere?According to the Copenhagen Interpretation, we can not directlyknow “the quantum world itself” ,but at least we can know itthrough observation. Is there a reason or a cause of ourphenomenal time? One of the keys to this issue (mystery?) is inmy opinion the “time operator”.


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Three types of time in quantum physics

Three types of time in quantum physics

(A) external time: is relevant to experiment.(B) observation time: is relevant to observation (detector)(C) internal time is relevant to the object itself.

・According to W.Heisenberg，here I generalize these definition (A),(B), (C) respectively as follows.

(A)　 is phenomenal time (classical time).(B)　 is the time of observation.(C)　 is non-phenomenal time (quantum time).

(B) is the so-called bridge between (A) and (C).Using terms of Aristotle(BC.384-322), (A) is in Energeia(Wirklichkeit) and (C) is in Dynamis (Moglichkeit). Therefore we canonly recognize “the time of (C)”, through the procedure (B).[T.Takabayashi, W.Heisenberg]


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History of the time operator

History of the time operator

It was well known that when we try to measure the energy ofparticles in a non-steady (non-stationary) state, the experimentalvalues are in dispersion.

So the uncertainty between the energy-measurement and the time ofobservation was considered. Eg. Heisenberg analyzed the experimentof “Stern-Gerlach” and he showed that δEδT∼ h, i.e. thetime-energy uncertainty relation

∆T∆E≥1

2~

where ∆ is the standard deviation.[M.Jammer]


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Next, Aharonov and Bohm constructed an operator with dimension oftime. This operator is named “Aharonov-Bohm time operator”.

T =1

2m(QP−1 + P−1Q)

Q is position operator, P is momentum operator.It is conjugate to the free particle Hamiltonian H = P2

2m. i.e.

Canonical Commutation Relations (CCR), [T ,H] = i . (~ = 1)

Is “T” a physical quantity? Is the spectrum is real? In other words,is T (essentially) self-adjoint?


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W.E.Pauli answered “No” to the above question.

This fact is easy to see from the “Von Neumann uniquenesstheorem”[cf, next page], i.e. that there is no self-adjoint operatorwhich satisfies CCR with the self-adjoint operator [Hamiltonian]bounded from below (or above).

[J.Muga, Jammer]

But, Pauli only indicated that for a (discrete) semi-boundedHamiltonian (self-adjont), there is no self-adjoint time operator.


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Theorem “Von Neumann uniqueness”

Let q and p be on the Hilbert space L2(R) defined by q := Mx ,p := −iDx , where Mx is the multiplication operator by variablex ∈ R, and Dx is the generalized differential operator in x . Thenself-adjoint pair (q, p) is a Weyl representation of the CCR, i,e.

e ispe itq = e−iste itpe isq

.(q, p) is called the Schrodinger representation of the CCR.

Let H be separable Hilbert space and (Q,P) is Weyl representationof the CCR. Then, if (Q,P) are irreducible, there exists a unitaryoperator U : H→L2(R) s.t, UQU−1 = q, UPU−1 = p.The spectrum is the following: σ(Q) = σ(q) = R, σ(P) = σ(p) = R————————————Therefore, if (T ,H) were a self-adjoint pair, because (T ,H) is aWeyl representation, σ(T ) = σ(q) = R, σ(H) = σ(p) = R.


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Classification of time operators according to CCR

Classification of time operators according to CCR

Nowadays we understand some types of time operators. TheAharonov-Bohm type is contained in this classification.

Now, we always suppose that Hamiltonian H is (essentially)self-adjoint (of course, we need to prove whether H is self-adjoint ornot.). If the time operator T is self-adjoint and H is unboundedself-adjoint, the Weyl representation exist. But here we would like toclassify time operators for general H .If we allow for T to be a symmetric operator, we can recognizeseveral types of T .

In the following, we introduce 4 types of time operators.


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(1)—[Definition “Weyl representation” type (1) ]

As we have already seen, Paul denied this type of time operator froma viewpoint of physical meaning. But in a mathematical meaning, weregard this type as a kind of time operator.Then self-adjoint pair (T ,H) is a Weyl representation of the CCR. i.e.

e isHe itT = e−iste itHe isT

. (T ,H) is one of Schrodinger representations of the CCR.

σ(T ) = R

σ(H) = R

.


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(2)—[Definition “weak Weyl representation” type (2) ]

A (closed) symmetric operator T on H is called a “weak Weylrepresentation”-type time operator w.r.t. a self-adjoint operator H onH if, for all t∈R, e−itHD(T ) ⊂ D(T ) and

Te−itHΨ = e−itH(T − t)Ψ

for all Ψ ∈ D(T ).[K.Schmudgen, M. Miyamoto]


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(3)—[Definition “inner product representation”-type (3) ]

A (closed) symmetric operator T on H is called an “inner productrepresentation”-type time operator w.r.t. a self-adjoint operator H onH if,

⟨TΨ,HΦ⟩ − ⟨HΨ,TΦ⟩ = i⟨Ψ,Φ⟩

for all Ψ,Φ ∈ D(H)∩D(T ), where ⟨a, b⟩ denotes the inner productof H.[A.Arai]


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(4)—[Definition “normal ”-type (4)]

A (closed) symmetric operator T on H is called a “normal”-typetime operator w.r.t. a self-adjoint operator H on H, if there issubspace D = 0, D⊂D(TH)∩D(HT ),

[T ,H]Ψ = iΨ

for all Ψ,Φ ∈ D.[A.Galapon]


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We know that type(1) ⊂ type(2) ⊂ type(4) ⊂ type(3).type(3) is the most general representation. The following theoremsconcerning on time operator of type (2) are known:

[Theorem -a]Let H be a self-adjoint and semi-bounded operator on H. Then thereis no self-adjoint time operator T of type (2) w.r.t. H which can beessentially self-adjoint. Therefore, in the case that H issemi-bounded, T is not observable (i.e, σ(T ) is not in R).

In other words, If T is essentially self-adjoint, then H and T are notsemi-bounded and

σ(H) = σ(T ) = R


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[Theorem -b]Suppose that a self-adjoint operator H has a time operator T of type(2) which is conjugate to H . Then H has only absolutely continuousspectrum. i.e. σ(H) = σac(H)

[Theorem -c]Let H be a self-adjoint operator on H and let T be a time operatorof type (2) w.r.t H . Then the following hold:(i) if H is bounded from below, then σ(T ) is either C orz ∈ C|ℑz > 0(ii) if H is bounded from above, then σ(T ) is either C orz ∈ C|ℑz < 0(iii) if H is bounded, then σ(T ) = C

[Arai, Miyamoto, Schmudgen, Reed and Simon]


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Examples of time operators according to the above classification

Examples of time operators according to the above

classification

type (1)Let be H as follows: H = aP (“a” is a real number). if weregard T = Q

aas time operator, then we can get it and this time

operator is self-adjoint.

type (1)(Freely falling particle [Busch]) Let be H as follows:H = P2

2m−mgQ (g is the gravity acceleration). Then we get

T = 1mg

P as self-adjoint time operator.


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type (2)The Aharonov-Bohm time operator is a type (2) operator w.r.t.H = P2

2m. (Of course, it is contained in type (4) and (3)).

type (2)The relativistic time operator T is type(1) w.r.t the freeHamiltonian for a relativistic quantum particle H =

√−∆+m2.

(m≥0) : T = HP−1Q + QP−1H (on a dense domain, usingFourier transformation. omitting details)

Further, we can get time operator w.r.t. the Dirac typeHamiltonian and we can generalize type (2) in Fock spaces. Andif in the above, these Hamiltonian have symmetric potential V(i.e. H + V ), we can get time operator under some conditions.(ex. using perturbation theory).


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type (4)Let H be self-adjont on H with purely discrete spectrum:σ(H) = σp(H) = En∞n=1 (En < En+1, limEn→∞ = ∞):and the multiplicity of each En being one and Σ∞

n=11E2n<∞.

Let en be the normalized eigenvector of H with eigenvalue En:Hen = Enen. ∥en∥ = 1.Then we can define a time operator T on (H):D(T ) := L(ek − el |k , l ∈ N), For all Ψ ∈ D(T ).TΨ := Σn =m

iEn−Em

⟨em,Ψ⟩en

This time operator is called “Galapon-time operator” and itbelongs to type (4) (therefore (3)), but this time operator doesnot belong to type (1), (2).And if infn,m∈N(En − Em) = 0, this time operator is essentiallyself-adjoint and bounded (In the case ofinfn,m∈N(En − Em) = 0, T is unbounded and we do not knownanything about T yet). [Galapon]


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[The example of Galapon-time operator (Harmonic oscillator [cf,POVM])]:

Let En = ω(n + 12), n ∈ 0 ∪N with a constant ω > 0.

In this case T is a bounded self-adjoint operator with D(T ) = H, and

TΨ = iωΣn=1(Σn =m

⟨em,Ψ⟩n−m

)en , Ψ ∈ Hwe can prove : σ(T ) = [−π

ω, πω]

As is well known, in the context of quantum physics, the sequenceω(n + 1/2)n=1 appears as the spectrum of the 1-dimensionalquantum harmonic oscillator H with mass m > 0H := P2

2m+ 1

2mω2Q2

[Galapon, G.Dorfmeister and J.Dorfmeister]


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Positive operator valued measure (POVM)

Positive operator valued measure (POVM)

These above methods are based on CCR representations : [T ,H] = i .Here we can look at another method. It is a method which is usingthe POVM. We regard POVM as generalized spectral measures(SPM).

[Definition]:Let BΩ be Borel-σ-field of sets on a compact Hausdorff space andB(H)+ := A∈B(H)|A ≥ 0. Then if a mapping F : BΩ → B(H)+satisfies the following conditions, F is called a POVM on Ω.

F (Ω) = I , F (∅) = 0

F (S) = F (S)∗ for all S ∈ BΩ

For mutually disjoint element of Sn∞n−1 ⊂ BΩ,F (∪∞

n=1Sn) = Σ∞n=1F (Sn), (strong-convergence).


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When F is POVM with F = F n, F is a SPM. As we know, thespectral theorem is valid when the operators corresponding tospectral measures are self-adjoint, bounded-normal and unitaryoperators. But a POVM only corresponds to an (but not unique)unbounded symmetric operator.⟨ψ,Tϕ⟩ =

∫Rλd⟨ψ, F (λ)ϕ⟩, ψ ∈ H, ϕ ∈ D(T ).

where, D(T ) := ϕ ∈ H|∫Rλ2d⟨ϕ,F (λ)ϕ⟩ <∞

[Theorem: M. Naimark]For all POVM F on K, there is a Hilbert space H which contains Kand a SPM E which satisfies the following:PE (S)P = F (S), for all S ∈ BΩ

where P is projection : H → K.

In general, F is not a SPM, so there is no theorem like the “SpectralTheorem”.Therefore, F is not unique, so there is no one -to-one correspondencebetween a POMV and a symmetric operator T .


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Using POVM, we can get a time operator as follows:

In the case: F is SPM [P.Busch](freely falling particle )Let be H as follows: H = P2

2m−mgQ (g is the gravity

acceleration). Then we get T =∫λdF (−mgλ) = 1

mgP as

self-adjoint time operator.

(1-dimensional quantum harmonic oscillator)Let H be a self-adjoint harmonic oscillator: H = 1

2(P2 + Q2),

(m = ~ = 1). Introduce the annihilation operatorA = 1

2(Q + iP), which gives the number operator N = A∗A, then

H = N +1

2I

.


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Then, for t ∈ [0, 2π], we define POVM as follows:

F (S) = Σn,m≥01

2π

∫e i(n−m)tdt|n⟩⟨m|

(mod 2π)

T =

∫ 2π

0

tλdF (λ) = Σm =n≥01

i(n −m)|n⟩⟨m|+ πI


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Aharanov-Bohm type(A) [K. Fredenhagen, R. Brunetti, M. Hoge]

Let H be a free particle Hamiltonian H = p2

2m. Then we can

define a POVM.

F (S) =

√PQ

2πm

∫e it

P2−Q2

2m dt, PQ 0

0, else.

Then, T is Aharanov-Bohm type.

(B) [J.Kijowski, Busch]Of same Aharanov-Bohm type, there is another POVM F :

F (S) =1

2π

∫S

dt

(∣∣∣∣∫ ∞

0

dP

√P

me

itP2

m

∣∣∣∣2 + ∣∣∣∣∫ 0

−∞dP

√−P

me

itP2

m

∣∣∣∣2)

In addition to above, the 1- dimensional relativistic timeoperator case and cosmological case (on the Wheeler-De Wittequation) is known. [K. Fredenhagen, R. Brunetti, M. Hoge]


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Summary and future plans

Summary and future plans

We understand that the time operator has a close relation toenergy. [T ,H] = i .Therefore, when we think about time in(quantum) physics, we have to consider the relation with theenergy operator.

In general we can not use CCR [T ,H ] = i in curved space time,because there is no Hamiltonian on it. In this case, what kind ofmethods can we use?

What would it mean if the spectrum of a time operator real?

How can our cognition of the time operator help us tounderstand quantum time?


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Philosophical essay

Philosophical essay

We can not recognize quantum time itself. We can only understandit through macro-time, and observation-time. So according to theCopenhagen Interpretation of N. Bohr, the physical phenomenon(macro-time) is the only important object. If this were true, we cannot reach quantum time (micro-time) and there is no existence ofquantum world (i.e. Agnosticism(Agnostizismus) )[B. C van Fraassen ].

Of course I think it is not possible directly. But, I think it is indirectlypossible.

L.Wittgenstein said that we have no private-language, so when youthink something on your own, you can not help but think anythingusing public-language, i.e. public structure.


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When we consider something in the micro-world, we can not helpusing the public-structure of our macro-world. Because we can havea dialog about quantum world, what we can do and should do is onlythe following:

I thinkthat the classical, macro world is not the secondary, subsidiary worldof the quantum world,

that when we deduce from the micro world the macro world, weshould not forget that we induce from the macro world. In otherwords, we should remember that we are always observers in thismacro-world [cf, micro-macro duality: I.Ojima].

“When you look at your neighbor , he or she will change theirattitude or behavior”.


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　 Thank you very much for this opportunity and your kind attention.


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Quantum physics and the time operator - eonet.ne.jptfujimoto21/Desy2.pdf · Quantum physics and the time operator Tadashi FUJIMOTO, Dept.Philosophy Ryukoku Univ. Kyoto AQFT Seminar: