Superselection Rules from Measurement Theory Shogo Tanimura Department of Complex Systems Science Graduate School of Information Science Nagoya University Reference: arXiv 1112.5701
Superselection Rules from Measurement Theory
Shogo Tanimura Department of Complex Systems Science Graduate School of Information Science
Nagoya University
Reference: arXiv 1112.5701
My recent popular articles โParadox of Photonโ Nikkei Science, 2012 March โNew Uncertainty Relation,โ 2012 April โWhat is Measurable?,โ 2012 July
Read it!
Plan of this talk โข Introduction โข A toy model: momentum superselection rule โข Tool: von Neumannโs indirect measurement
model โข Basic notions: isolated conservation law,
covariant indicator โข Main theorem: we derive the superselection
rule from a conservation law in measurement process. 3
Superselection Rule โข ๐ฝ: superselection charge โข ๐ด : self-adjoint operator, ๐ดโ = ๐ด The superselection rule states ๐ด is measurable โ ๐ด, ๐ฝ = 0 By contraposition, ๐ด, ๐ฝ โ 0 โ ๐ด is non-measurable
The superselection rule is a necessary condition for a self-adjoint operator ๐ด to be a measurable quantity.
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History of superselection rule Wick, Wigner, Wightman (1952) noticed that not every self-adjoint operator represents a physically measurable quantity. ๐๏ผ Dirac field operator These are self-adjoint but they are not measurable. Charge density and current density are measurable. Although the intensity of electron wave is measurable, its phase is non-measurable.
12๐ + ๐โ , 1
2๐๐ โ ๐โ
๐โ ๐, ๐๏ฟฝ๐พ๐๐
We can observe an interference fringe of the electron wave but we cannot determine its phase.
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Univalence superselection rule
๐ฝ = ๐ 2๐ : rotation by 360 degree around any axis. A measurable quantity ๐ด must satisfy On the other hand, the Dirac spinor field ๐ satisfies Thus the Dirac spinor field ๐ itself is not a measurable quantity even though ๐โ ๐ is measurable.
๐ 2๐ โ ๐ด๐ 2๐ = ๐ด or equivalently, ๐ด, ๐ฝ = 0
๐ 2๐ โ ๐๐ 2๐ = โ๐, ๐ 2๐ โ ๐โ ๐ 2๐ = โ๐โ
another derivation by Hegerfeldt, Kraus, Wigner (1968)
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How did they notice it? Around 1950, physicists discussed definition of the parity transformation of the Dirac field. It was not uniquely defined but it had an ambiguity. Parity transform: The phase factor is not uniquely determined. In 1952, WWW noted that the parity transformation of the Dirac spinor is allowed to be unfixed since the Dirac spinor itself is non-measurable.
๐ ๐, ๐ก โ ๐ฑ๐ ๐, ๐ก = ๐๐๐๐พ0๐ โ๐, ๐ก
๐๐๐
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Correspondence of mathematical notion to physical observable
โข Mathematical notion: self-adjoint operator โข Physical notion: observable (measurable quantity) Do they have one-to-one correspondence? In the usual framework of quantum mechanics, their one-to-one correspondence is assumed.
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von Neumannโs argument (1932) After showing that every observable is representable by a self-adjoint operator, von Neumann argued that it is appropriate to assume that there is a physical observable corresponding to each self-adjoint operator. observable โ (โ?) self-adjoint: ๐ดโ = ๐ด The superselection rule tells that this assumption is not appropriate. There is a self-adjoint operator that does NOT correspond to any physical observable.
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von Neumann, โMathematical Foundations of Quantum Mechanics,โ Section IV.2
โข ไบไธใปๅบ้ใปๆ่ค ่จณ๏ผ1957ๅนด๏ผ p.250 ใ้ๅญๅๅญฆ็็ณปใฎ็ฉ็้ใซๅฏพใใฆ่ถ ๆฅตๅคงใชใจใซใใผใไฝ็จ็ด ใไธๆ็ใซๅฏพๅฟใใใใใใใจใฏ๏ผๆใ ใฎ็ฅใฃใฆใใ้ใใงใใใ๏ผใใใซๅ ใใฆ๏ผใใใใฎๅฏพๅฟใฏไธๅฏพไธใงใใ๏ผใใชใใก๏ผใในใฆใฎ่ถ ๆฅตๅคงใจใซใใผใไฝ็จ็ด ใฏ็พๅฎใซ็ฉ็้ใซๅฏพๅฟใใฆใใ๏ผใจไปฎๅฎใใใฎใ้ฝๅใใใ๏ผใ
โข Original German expression by von Neumann (1932) ใ... es ist zweckmรคssig anzunehmen, ...ใ
โข English translation by Beyer (1955) ใ... it is convenient to assume, ...ใ
โข English translation by Wightman (1995) ใ... it is appropriate to assume, ...ใ
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Other examples: Not every self-adjoint operator
corresponds to observable
โข Lorentz boost generators โข Dilatation generator
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For making a clear argument we need a clear definition
โข self-adjoint: mathematically well-defined notion
โข observable (something measurable): not clear
It is necessary to formulate the notion of measurement.
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Toy model to help understanding of the superselection rule
โข ๐ particles in one-dimensional space โข masses: ๐1,๐2, โฆ ,๐๐ โข positions: ๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐ โข momenta: ๐1,๐2, โฆ , ๐๐ โข An apparatus has a meter observable ๐. โข Time-evolution of the whole system is
described by a unitary operator ๐ ๐
๐ฅ1 ๐ฅ2 ๐ฅ3 ๐ฅ4 ๐ฅ5 13
Indirect measurement model โข The object system (๐-particle system) has a Hilbert space โ,
while the apparatus has a Hilbert space ๐. โข The initial state of the whole system is ๐โจ๐ โ โโจ๐ โข An observable ๐ด to be measured is a self-adjoint operator on โ, while a meter observable ๐ is a self-adjoint operator on ๐.
โข Interaction between them is described by a unitary operator ๐ = ๐โ๐๐๐/โ on the composite system โโจ๐.
โข The meter ๐โ ๐๐ is read out by means of the Born probability rule.
Object System Apparatus (observing system)
meter ๐โถ ๐โ ๐๐ observables ๐ด,๐ต ๐ initial state ๐ โ โ initial state ๐ โ ๐
interaction
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โข Suppose that we want to measure the position of the center of mass of the particles:
๐ด = ๐ โโ ๐๐๐ฅ๐๐๐=1โ ๐๐๐๐=1
โข Assume that the total momentum of the particles is conserved during the measurement process (isolated conservation law):
๐ฝ = ๐ โ๏ฟฝ ๐๐๐
๐=1, ๐โ ๐๐ = ๐
โข Does the meter move as ๐โ ๐๐ = ๐ + ๐ ? โข Answer: It is impossible.
Requirement
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The total momentum of the particles ๐ = ๐โจ1 and the meter position operator ๐ = 1โจ๐ on โโจ๐ commute
๐,๐ = 0 Since the time-evolution acts as an automorphim,
๐โ ๐๐,๐โ ๐๐ = ๐โ ๐,๐ ๐ = 0 On the other hand, the momentum conservation and the meter shift condition imply
๐โ ๐๐,๐โ ๐๐ = ๐,๐ + ๐ = ๐,๐ = โ๐โ These give a contradiction. In general, a quantity ๐ด measurable in the sense ๐โ ๐๐ = ๐ + ๐ด must satisfy ๐,๐ด = 0.
Proof
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General scheme which gives a rise of superselection rule
Object System Apparatus (observing system)
covariant meter ๐โถ ๐โ ๐๐ = ๐ + ๐ด
observable ๐ด
isolated conserved quantity ๐ฝ โถ ๐โ ๐ฝ๐ = ๐ฝ
๐
interaction
Derivation of the superselection rule
๐ฝ,๐ = 0, 0 = ๐โ ๐ฝ,๐ ๐
= ๐โ ๐ฝ๐,๐โ ๐๐= ๐ฝ,๐ + ๐ด= ๐ฝ,๐ด
โด ๐ฝ,๐ด = 0. 17
definition of measurability
A more general proof is given in my paper.
Measurability โข Physically meaningful measurement requires
covariance between the quantity to be measured and the quantity to be read out.
โข But the structure of interaction between the object system and the apparatus may or may not allow the covariance.
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: quantity to be read
: quantity to be known
meter ๐
human weight ๐ด
meter ๐ โ ๐๐ weight ๐ด โ ๐ด๐
covariance is necessary for making meaningful measurement.
The superselection rule associated to the momentum conservation law demands any measurable quantity ๐ด must satisfy ๐,๐ด = 0, where ๐ = โ ๐๐๐ is the total momentum. Measurable: relative coordinates
๐ด = ๐ฅ๐ โ ๐ฅ๐ , ๐ด =๐1๐ฅ1 + ๐2๐ฅ2๐1 + ๐2
โ ๐ฅ3
Non-measurable: absolute coordinates of the particle
๐ด = ๐ฅ๐ , ๐ด =โ ๐๐๐ฅ๐๐๐=1โ ๐๐๐๐=1
Measurable/non-measurable quantities
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Symmetries / Superselection charges
โข 2ฯ rotation invariance / univalence The Dirac spinor, which is not invariant under 2ฯ rotation, is non-measurable.
โข U(1) invariance / electric charge, baryon number The phases of matter waves of electron or neutron are non-measurable.
โข Notice: Photon number is not conserved, hence, the phase of electromagnetic wave is measurable.
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Paradox associated with Non-abelian symmetry
For example, SO(3) rotation invariance implies the conservation of angular momenta ๐ฝ๐ฅ, ๐ฝ๐ฆ , ๐ฝ๐ง, which are noncommutative each other. Hence the superselection rule prohibits the measurements of angular momenta. But, in actual experiments, we measure the spin angular momenta of electrons or photons. How is it possible? 21
Solution of the angular momentum paradox
The SO(3) rotation invariance is broken by introducing external magnetic field (Zeeman effect or Stern-Gerlach setting) for nuclei or electrons, polarization filter or birefringent crystal for photons. All of the measurements of angular momenta introduce a coupling of the object system and the apparatus that breaks the isolation of the system and allows exchange of angular momenta between the two systems.
๐ป = ๐๐บ โ ๐ฉ [๐ป,๐บ] โ 0 ๐
๐ฉ ๐ป = ๐๐บ โ ๐ณ ๐ป,๐บ โ 0
๐ป,๐บ + ๐ณ = 0
object spin external field
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Why we can measure rotationally non-invariant quantities?
If the SO(3) rotation invariance is preserved within a microscopic system, we cannot measure any rotationally variant quantity from outside. However, actually we can measure it since the rotation invariance is spontaneously broken at the macroscopic scale. We can construct apparatus which has a non-spherical shape. Thus, we can apply rotationally non-invariant external field on a microscopic system.
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How can we overcome the U(1) superselection rule?
The isolated conservation of the U(1) charge makes measurements of gauge variant quantities impossible. By breaking the isolation, we can make such a measurement possible. Example: superconductivity, Josephson junction
๐1 = ๐1 ๐๐๐1
๐2 = ๐2 ๐๐๐2
๐ป = ๐๐1โ ๐2 + ๐๐2โ ๐1
๐ผ = ๐1โ ๐2 โ ๐2โ ๐1
isolation is broken.
Superconductor
Cooper condensate ๐ = ๐ ๐๐๐
superselection rule prevents the measurement.
๐ผ
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Why the SU(3) color is invisible?
In QCD, colored quantities like quark and gluon fields are non-measurable from outside of hadrons. Moreover, since SU(3) is non-abelian, color charge itself is non-measurable. If there are objects in which the color symmetry are spontaneously broken, we can measure their relative color difference.
๐1๐1 Color superconductor
๐๐
superselection rule prevents the measurement.
๐2๐2
isolation is broken. 25
Uncertainty relation under a conservation law: Wigner-Araki-Yanase-Ozawa theorem
๐ด : observable to be measured ๐ : meter of the apparatus ๐ฝ1 : quantity of the object system ๐ฝ2 : quantity of the apparatus ๐ : unitary time-evolution ๐บ = ๐โ ๐ : initial state of the whole system
๐ ๐ด 2 = ๐บ ๐โ ๐๐ โ ๐ด 2 ๐บ : measurement error
๐ ๐ฝ1 2 = ๐บ ๐ฝ1 2 ๐บ โ ๐บ ๐ฝ1 ๐บ 2 : standard deviation Assume ๐โ ๐ฝ1 + ๐ฝ2 ๐ = ๐ฝ1 + ๐ฝ2
๐ ๐ด 2 โฅ๐ด, ๐ฝ1 2
4 ๐ ๐ฝ1 2 + ๐ ๐ฝ2 2 2 Theorem:
interaction
Object Apparatus
๐, ๐ฝ2, ๐ ๐ด, ๐ฝ1,๐ ๐ด, ๐ฝ1 โ 0
๐
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Comparison of the WAY-Ozawa theorem with the superselection rule
The WAY theorem: The conservation of total charge ๐โ ๐ฝ1 + ๐ฝ2 ๐ = ๐ฝ1 + ๐ฝ2 implies that a non-vanishing error of the measurement ๐ ๐ด is evitable.
Superselection rule: The isolated conservation law ๐โ ๐ฝ1๐ = ๐ฝ1 implies the impossibility of ๐โถ ๐โ ๐๐ = ๐ + ๐ด, that is, the meter ๐ cannot move covariantly to the object quantity ๐ด. This is an extremal form of the error-disturbance uncertainty relation.
๐
interaction
Object Apparatus ๐ด, ๐ฝ1,๐ ๐ด, ๐ฝ1 โ 0 ๐, ๐ฝ2, ๐
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Accessible and Inaccessible Levels of the World Macroscopic level
Microscopic level
Invisible quantity ๐ต such that [๐ต, ๐ฝ] โ 0
Visible quantity ๐ด must satisfy [๐ด, ๐ฝ] = 0 barrier by the
isolated conservation law
quantity ๐ด
Less isolated conservation laws open the window of measurement and control
symmetry breaking at macroscopic scale
More visible quantities
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Hierarchical structure of the nature
Quarks and Gluons
Hadrons and Nuclei Atoms and Molecules
color symmetry is kept chiral symmetry is broken
rotation symmetry is broken
a picture inspired by P. W. Andersonโs โMore is differentโ
An isolated conservation law, that is, a superselection rule defines a border of two hierarchy, beyond which the underlying entities cannot be seen from the overlying entities. On the other hand, spontaneous symmetry breaking provides a scope or a handle with which we can observe or control the underlying level from the overlying level.
Underlying structure
Life, Society, Planets, Cosmos
Condensed matter, Crystal, Cell
translation symmetry is broken
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Conclusion โข The uncertainty relation tells that we cannot precisely
measure a quantity A without disturbing another quantity J such that [A,J ] โ 0.
โข If the disturbance of J is absolutely prohibited, namely, if the object system has the isolated conserved quantity J, the measurement of A becomes impossible. This is the superselection rule.
โข The superselection rule is understood as a consequence of symmetry from a viewpoint of measurement theory.
โข We can overcome the superselection rule by introducing explicit or spontaneous symmetry breakings.
โข Isolated conservation laws and spontaneous symmetry breakings build the hierarchical structure of the nature.
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The End
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