tensor categorical structure of observables in โฆtanimura/lectures/tanimura...dimensional analysis...
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Tensor categorical structure of observables in classical physics
Shogo TanimuraDepartment of Complex Systems Science
Graduate School of InformaticsNagoya University
Symposium on the Categorical Unity of the SciencesKyoto University, 2019 March
Introduction of myself
โข My name is Shogo Tanimura..โข I am a theoretical physicist.โข My main concerns are foundation of
quantum theory, dynamical system theory, application of differential geometry to physics, and category theory.
โข Todayโs topic is a rather primitive application of category theory to both classical and quantum physics.
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Plan of this talk
1. Review of elementary dimensional analysis
2. Category of physical quantities3. Dimensional analysis is formulated as
tensor category4. Unit system is a functor5. Unit transformation law is a natural
transformation6. Observable algebra in quantum
physics in terms of category3
Dimensional analysis
Calculus of physical quantities
3kg + 500g = 3000g + 500g = 3500g
4m + 70cm = 4m + 0.7m = 4.7m
40kg + 8cm = 48 ? illegal !
Addition, equality and inequality of quantities must be homogeneous with respect to physical dimension.
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NONSENSE Sum
Fredrick I. Olness at Snowmass Villagehttp://www.physics.smu.edu//~olness/www/index.html
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Dimensional analysisin physics
[kg] = Mass[meter] = Length[sec] = Time[pressure] = [N/m2] = MLT-2L-2 = ML-1T-2
[mass density] = ML-3
[pressure]----------------- = L2 T-2[density]
[velocity] = L1 T-1 = [๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ][๐๐๐ฉ๐ฉ๐๐๐ฉ๐ฉ๐๐๐๐๐๐]
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Dimensional analysisin physics
Estimation of the sonic speed in airRelevant parameters:
air pressure = 1 atm = 1.013 x 105 N/m2
air density = 1.2kg/m3
[pressure]----------------- = L2 T-2[density]
[velocity] = L1 T-1 = [๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ][๐๐๐ฉ๐ฉ๐๐๐ฉ๐ฉ๐๐๐๐๐๐]
= 290m/s
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Dimensional analysisin physics
Physicists feel uneasy with dimensionally incoherent formula :
๐ธ๐ธ = ๐๐๐๐2 + 12๐๐๐ฃ๐ฃ
2 + 14๐๐๐ฃ๐ฃ
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Physicist notice this formula is wrong by a glance.
Correct formula:
๐ธ๐ธ = ๐๐๐๐2 1 + 12๐ฃ๐ฃ๐๐2 + 3
8๐ฃ๐ฃ๐๐4 + โฏ =
๐๐๐๐2
1 โ ๐ฃ๐ฃ๐๐2
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Dimensional analysisin mathematics
Quadratic equation and its solution:๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ = 0
๐ฅ๐ฅ =โ๐๐ ยฑ ๐๐2 โ 4๐๐๐๐
2๐๐Requiring homogeneity of dimensions:
๐ด๐ด๐๐2 = ๐๐๐ฅ๐ฅ2 = ๐๐๐ฅ๐ฅ = [๐๐]๐๐ = ๐ด๐ด, ๐๐ = ๐ด๐ด๐๐, ๐๐ = ๐ด๐ด๐๐2
๐ฅ๐ฅ =โ๐๐ ยฑ ๐๐2 โ 4๐๐๐๐
2๐๐=๐ด๐ด๐๐๐ด๐ด
= ๐๐
Consistent !9
Dimensional analysisin mathematics
Indefinite integral:
๏ฟฝ๐๐
๐๐2 + ๐ฅ๐ฅ2๐๐๐ฅ๐ฅ = tanโ1
๐ฅ๐ฅ๐๐
Homogeneity of dimensions is kept.
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Dimensional analysisin quantum physics
Hamiltonian of harmonic oscillator:๏ฟฝ๐ป๐ป = 1
2๐๐๏ฟฝฬ๏ฟฝ๐2 + 1
2๐๐๐๐2 ๏ฟฝ๐ฅ๐ฅ2 + ๐๐๐๐๏ฟฝ๐ฅ๐ฅ
๏ฟฝ๐ป๐ป| โฉ๐๐๐๐ = ๐ธ๐ธ๐๐| โฉ๐๐๐๐
๐ธ๐ธ๐๐ = โ๐๐ ๐๐ + 12 โ
๐๐๐๐2
2๐๐2
๐๐๐๐ ๐ฅ๐ฅ =1
2๐๐ 1/4ฮ๐ฅ๐ฅ1/2 ๐ป๐ป๐๐๐ฅ๐ฅฮ๐ฅ๐ฅ ๐๐โ
๐ฅ๐ฅ24ฮ๐ฅ๐ฅ2
ฮ๐ฅ๐ฅ2 โโ๐๐
2๐๐๐๐2Homogeneity of dimensions is kept again.
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Question
Dimensional analysis works also in quantum theory.However, the standard mathematical formulations of quantum theory (Hilbert space formalism, von Neumann algebra, C*-algebra) do not concern physical dimensions of observables.I would like to formulate quantum theory in a language that involves physical dimensions.
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A Category of Simple Quantities
โข Object: Simple quantity = 1-dim vector spaceโ additionโ scalar multiplication by real number
โข Arrow: Linear mappingVelocity Hom ๐ฟ๐ฟ;๐๐ = ๐ฟ๐ฟโจ๐๐โ
Time ๐๐ โ ๐ฟ๐ฟ Length, distance
๐ก๐ก โผ ๐๐ = ๐ฃ๐ฃ๐ก๐ก๐ฃ๐ฃ = 36 km/h = 10m/s
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A Category of Simple Quantities
โข Composition of linear mappings
๐ฃ๐ฃ = 36 km/h = 10m/s
๐๐ = 100yen/300m = 13
yen/m
๐๐ โ ๐ฃ๐ฃ = 103
yen/s = 3.33yen/s
Velocity ๐ก๐ก โผ ๐๐ = ๐ฃ๐ฃ๐ก๐ก
Time ๐๐ โ ๐ฟ๐ฟ Length, distance
โ โ Meter ๐๐ โผ ๐๐ = ๐๐๐๐
๐ถ๐ถ Cost
๐๐
๐ฃ๐ฃ
๐๐ โ ๐ฃ๐ฃ
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A Category of Simple Quantities
โข Hom-set is also a quantity๐๐ = Hom ๐๐; ๐ฟ๐ฟ โ ๐ฃ๐ฃ, ๐๐๐ฃ๐ฃ, ๐ฃ๐ฃ1 + ๐ฃ๐ฃ2
โข Dual quantity: ๐๐โ = Hom(๐๐;โ)โ frequency, density
โข Multilinear mappings define tensor products๐๐โโจ๐๐โ โ Hom(๐๐,๐๐;โ) โ Hom(๐๐โจ๐๐;โ)
๐๐ ร ๐๐ โ ๐๐โจ๐๐
โ โ โ!๐๐โ๐๐
โ๐๐
โจUniversality
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A Category of Simple Quantities
โข The set of all endomorphisms of 1-dim vector space is isomorphic to real number field: dimensionless quantity
Hom ๐ฟ๐ฟ; ๐ฟ๐ฟ โ Hom ๐๐;๐๐ โ โโข Unit and Coefficient:
๐๐ โ Hom โ;๐๐๐๐ โผ (๐ธ๐ธ๐๐ โถ ๐๐ โผ ๐๐๐๐)
kg โผ (๐ธ๐ธkg โถ 60 โผ 60kg)If ๐๐ โ ๐๐, ๐ธ๐ธ๐๐:โโถ ๐๐ is invertible:
๐ฉ๐ฉ๐๐ = ๐ธ๐ธ๐๐ โ1:๐๐ โถ โ16
Higher-Dimensional Quantities
โข Object: quantity = finite dimensional vector space over โ
โข Arrow: linear mapping
โข simple quantity: pressure
โข higher-dim quantity: tension
area ๐ฟ๐ฟโจ๐ฟ๐ฟ โ ๐น๐น force๐๐
area ๐๐โจ๐๐ โ ๐น๐น force๐๐
dim ๐๐ = dim๐น๐น = 3,17
Higher-Dimensional Quantities
โข Multilinear mappings๐๐1 ร ๐๐2 ร โฏร ๐๐๐๐ โถ ๐๐๐๐ ร ๐๐ ร โฏร ๐๐ โถ ๐๐
โข symmetric bilinear โ inner productโข antisymmetric โ exterior productโข orientation โ parity, twisted quantityโข contraction, trace
๐๐โ ร ๐๐ โ ๐๐โโจ๐๐โ โ
โpairing
โจ
contraction
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How to make equivalence classes of quantities?
When we have transformation laws among measurable quatities but no general measure
salt by cup
sugar by spoon
milk by bottle
coffee by cup
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Abstractization of Quantities
completed category of quantities
category of bare quantities
Limitweight of salt
weight of sugar
weight of milk
weight of coffee
salt by cup
sugar by spoon
milk by bottle
coffee by cup
abstract weight
embedding functor
gravity
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Abstractization of Quantities
category of abstract quantities
category of bare quantities
salt by cup
sugar by spoon
milk by bottle
coffee by cup
weight
forget, abstractize
force
length area volume
energy
acceleration
stage of dimensional analysis
time
velocity
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Unit system is a functor,Unit transformation is a natural transformation
category of abstract quantities
category of real number vector spaces
time โ length๐ฃ๐ฃ
carmile-hour functor
SI functor
2 hours โ 50 miles๐ฃ๐ฃ = 25 mile/h
7,200 sec โ 80,000 m
๐ฃ๐ฃ = 11. 1m/s
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Fineness of classification
fine category of abstract quantities
coarse category of abstract quantities
๐ธ๐ธ = ๐๐๐๐2
natural unit functor
mass
energy
frequency
momentum
lengthโ
๐ธ๐ธ = โ๐๐
๐ธ๐ธ = ๐๐๐๐
๐๐ = โ๐๐ =โ๐๐
mass
energy
frequency
momentum
lengthโ
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How to quantize?= How to make non-commutative?
โข Tensor product of classical quantities is symmetric:
๐๐โจ๐๐ โ ๐๐โจ๐๐
โข In quantum theory, we would like to introduce non-commutative products of observables.
โข Algebraic structure must be consistent with dimensional analysis:
๐ธ๐ธ =12๐๐
๐๐2 +12๐๐๐๐2๐ฅ๐ฅ2 + ๐๐๐๐๐ฅ๐ฅ
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Toward quantization
category of classical physical quantities
๐ฟ๐ฟ
๐๐ = ๐๐โจ๐ฟ๐ฟโจ๐๐โ1๐ฟ๐ฟโจ๐ฟ๐ฟ
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category of quantum observables
โ โ โ โ โ ๐ฟ๐ฟ โ๐๐
๐๐โจ๐ฟ๐ฟ
๐ฟ๐ฟโ = ๐ฟ๐ฟโ1๐๐
๐ธ๐ธ = ๐๐โจ๐ฟ๐ฟ2โจ๐๐โ2
1
๐ฅ๐ฅ + ๐ฆ๐ฆ
๐ฆ๐ฆโ โ ๐ฟ๐ฟ โ
โ โ ๐ฟ๐ฟ โ
๐ฅ๐ฅ
Hom ๐ด๐ด โ,๐ต๐ต โ isrequired to be a vector space.
โ โ ๐๐ โ๐๐๐ฅ๐ฅ
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Category of quantum observables
object: asterisk marked by classical quantityarrow: quantum observableEach hom-set is a vector space over โ.composition: product of quantum observables
โ โ ๐ฟ๐ฟ โ๐ฅ๐ฅ
๐๐๐ฅ๐ฅ๐๐โจ๐ฟ๐ฟ โโ โ๐๐๐ฅ๐ฅ โ ๐ฅ๐ฅ
๐ฅ๐ฅ
โ โ ๐ฟ๐ฟ โ๐ฅ๐ฅ
๐๐๐ฅ๐ฅ๐๐ โ โ ๐๐โจ๐ฟ๐ฟ โ
โโ๐๐๐ฅ๐ฅ
๐๐๐ฅ๐ฅ โ ๐ฅ๐ฅ โ ๐ฅ๐ฅ โ ๐๐๐ฅ๐ฅ26
Algebraic equation of quantum observables
Description of harmonic oscillator hamiltonian
โ โ ๐ฟ๐ฟ โโ ๐ฟ๐ฟ2 โ โ ๐๐โจ๐ฟ๐ฟ2โจ๐๐โ2 โ๐ฅ๐ฅ
๐๐๐ฅ๐ฅโ
๐ฅ๐ฅ12๐๐๐๐
2
๐๐ โ โ ๐๐2 โ โ ๐๐โ1โจ๐๐2 โ๐๐๐ฅ๐ฅ 1
2๐๐
๐ป๐ป =12๐๐
๐๐๐ฅ๐ฅ2 +12๐๐๐๐2๐ฅ๐ฅ2
โ ๐๐โจ๐ฟ๐ฟ2โจ๐๐โ2 โ27
Representation functor sends objects to Hilbert spaces and observables to operators
Category of Hilbert spaces with physical-quantity coefficients
๏ฟฝ๐ป๐ป =12๐๐
๏ฟฝฬ๏ฟฝ๐๐ฅ๐ฅ2 +12๐๐๐๐2 ๏ฟฝ๐ฅ๐ฅ2
โ ๐๐โจ๐ฟ๐ฟ2โจ๐๐โ2โจโ
โ โ ๐ฟ๐ฟโจโ โ ๐ฟ๐ฟ2โจโ โ ๐๐โจ๐ฟ๐ฟ2โจ๐๐โ2โจโ๏ฟฝ๐ฅ๐ฅ
๏ฟฝ๐๐๐ฅ๐ฅ โ
๏ฟฝ๐ฅ๐ฅ12๐๐๐๐
2๏ฟฝ1
๐๐โจโ โ ๐๐2โจโ โ ๐๐โ1โจ๐๐2โจโ12๐๐๏ฟฝ1๏ฟฝ๐๐๐ฅ๐ฅ
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Eigenvector subspace is an equalizer
Category of Hilbert spaces with physical-quantity coefficients.Eigenvalue problem: ๏ฟฝ๐ป๐ป| โฉ๐๐ = ๐๐| โฉ๐๐
๏ฟฝ๐ป๐ปโ โ ๐ธ๐ธโจโ
๐๐๏ฟฝ1
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Summary 1/3
1. Quantities in classical physics forms a tensor category.
2. A set of isomophic quantities define an abstract quantity.
3. Functor sends a fine category to a coarse category of quantities.
4. Unit system is a functor from quantities to real values.
5. Unit transformation law is a natural transformation.
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Summary 2/3
6. An observable in quantum physics is an arrow that multiplies physical quantity on an object:
โ โ ๐ฟ๐ฟ โ ๐๐ โ โ ๐ฟ๐ฟโจ๐๐ โ7. Hom-set is a vector space over โ.8. This structure guarantees
homogeneity of physical dimensions of terms in equations of observables.
๐ฅ๐ฅ ๐ฅ๐ฅ
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Summary 3/3
9. Composition of quantum observables is non-commutative in general.
10. Representation functor sends objects to Hilbert spaces with physical-quantity coefficients and arrows to operators:
โ โ ๐ฟ๐ฟโจโ ๐๐โ โ ๐ฟ๐ฟโจ๐๐โจโ11. Spectral values of observable
operators have suitable physical dimensions.
๏ฟฝ๐ฅ๐ฅ ๏ฟฝ๐ฅ๐ฅ
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References
1. Tanimura, โTopology, category, and differential geometry: from the viewpoint of dualityโ (in Japanese)
2. Tanimura, Series of lectures on geometry and physics published in Mathematical Science (in Japanese)
3. Kitano, โMathematical structure of unit systemsโ J. Math. Phys. 54, 052901 (2013)
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Thank you for your attention
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