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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
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Introduction of myself
β’ My name is Shogo Tanimura..β’ I am a theoretical physicist.β’ My main concerns are foundation of
quantum theory, dynamical system theory, application of differential geometry to physics, and category theory.
β’ Todayβs 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
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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 = [π©π©π©π©π©π©π©π©π©π©π©π©π©π©π©π©][πππ©π©πππ©π©ππππππ]
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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 = [π©π©π©π©π©π©π©π©π©π©π©π©π©π©π©π©][πππ©π©πππ©π©ππππππ]
= 290m/s
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Dimensional analysisin physics
Physicists feel uneasy with dimensionally incoherent formula :
πΈπΈ = ππππ2 + 12πππ£π£
2 + 14πππ£π£
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Physicist notice this formula is wrong by a glance.
Correct formula:
πΈπΈ = ππππ2 1 + 12π£π£ππ2 + 3
8π£π£ππ4 + β― =
ππππ2
1 β π£π£ππ2
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Dimensional analysisin mathematics
Quadratic equation and its solution:πππ₯π₯2 + πππ₯π₯ + ππ = 0
π₯π₯ =βππ Β± ππ2 β 4ππππ
2ππRequiring homogeneity of dimensions:
π΄π΄ππ2 = πππ₯π₯2 = πππ₯π₯ = [ππ]ππ = π΄π΄, ππ = π΄π΄ππ, ππ = π΄π΄ππ2
π₯π₯ =βππ Β± ππ2 β 4ππππ
2ππ=π΄π΄πππ΄π΄
= ππ
Consistent !9
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Dimensional analysisin mathematics
Indefinite integral:
οΏ½ππ
ππ2 + π₯π₯2πππ₯π₯ = tanβ1
π₯π₯ππ
Homogeneity of dimensions is kept.
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Dimensional analysisin quantum physics
Hamiltonian of harmonic oscillator:οΏ½π»π» = 1
2πποΏ½ΜοΏ½π2 + 1
2ππππ2 οΏ½π₯π₯2 + πππποΏ½π₯π₯
οΏ½π»π»| β©ππππ = πΈπΈππ| β©ππππ
πΈπΈππ = βππ ππ + 12 β
ππππ2
2ππ2
ππππ π₯π₯ =1
2ππ 1/4Ξπ₯π₯1/2 π»π»πππ₯π₯Ξπ₯π₯ ππβ
π₯π₯24Ξπ₯π₯2
Ξπ₯π₯2 ββππ
2ππππ2Homogeneity 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
β’ Object: Simple quantity = 1-dim vector spaceβ additionβ scalar multiplication by real number
β’ Arrow: Linear mappingVelocity Hom πΏπΏ;ππ = πΏπΏβ¨ππβ
Time ππ β πΏπΏ Length, distance
π‘π‘ βΌ ππ = π£π£π‘π‘π£π£ = 36 km/h = 10m/s
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A Category of Simple Quantities
β’ Composition of linear mappings
π£π£ = 36 km/h = 10m/s
ππ = 100yen/300m = 13
yen/m
ππ β π£π£ = 103
yen/s = 3.33yen/s
Velocity π‘π‘ βΌ ππ = π£π£π‘π‘
Time ππ β πΏπΏ Length, distance
β β Meter ππ βΌ ππ = ππππ
πΆπΆ Cost
ππ
π£π£
ππ β π£π£
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A Category of Simple Quantities
β’ Hom-set is also a quantityππ = Hom ππ; πΏπΏ β π£π£, πππ£π£, π£π£1 + π£π£2
β’ Dual quantity: ππβ = Hom(ππ;β)β frequency, density
β’ Multilinear mappings define tensor productsππββ¨ππβ β Hom(ππ,ππ;β) β Hom(ππβ¨ππ;β)
ππ Γ ππ β ππβ¨ππ
β β β!ππβππ
βππ
β¨Universality
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A Category of Simple Quantities
β’ The set of all endomorphisms of 1-dim vector space is isomorphic to real number field: dimensionless quantity
Hom πΏπΏ; πΏπΏ β Hom ππ;ππ β ββ’ Unit and Coefficient:
ππ β Hom β;ππππ βΌ (πΈπΈππ βΆ ππ βΌ ππππ)
kg βΌ (πΈπΈkg βΆ 60 βΌ 60kg)If ππ β ππ, πΈπΈππ:ββΆ ππ is invertible:
π©π©ππ = πΈπΈππ β1:ππ βΆ β16
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Higher-Dimensional Quantities
β’ Object: quantity = finite dimensional vector space over β
β’ Arrow: linear mapping
β’ simple quantity: pressure
β’ higher-dim quantity: tension
area πΏπΏβ¨πΏπΏ β πΉπΉ forceππ
area ππβ¨ππ β πΉπΉ forceππ
dim ππ = dimπΉπΉ = 3,17
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Higher-Dimensional Quantities
β’ Multilinear mappingsππ1 Γ ππ2 Γ β―Γ ππππ βΆ ππππ Γ ππ Γ β―Γ ππ βΆ ππ
β’ symmetric bilinear β inner productβ’ antisymmetric β exterior productβ’ orientation β parity, twisted quantityβ’ contraction, trace
ππβ Γ ππ β ππββ¨ππβ β
βpairing
β¨
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 β lengthπ£π£
carmile-hour functor
SI functor
2 hours β 50 milesπ£π£ = 25 mile/h
7,200 sec β 80,000 m
π£π£ = 11. 1m/s
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Fineness of classification
fine category of abstract quantities
coarse category of abstract quantities
πΈπΈ = ππππ2
natural unit functor
mass
energy
frequency
momentum
lengthβ
πΈπΈ = βππ
πΈπΈ = ππππ
ππ = βππ =βππ
mass
energy
frequency
momentum
lengthβ
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How to quantize?= How to make non-commutative?
β’ Tensor product of classical quantities is symmetric:
ππβ¨ππ β ππβ¨ππ
β’ In quantum theory, we would like to introduce non-commutative products of observables.
β’ Algebraic structure must be consistent with dimensional analysis:
πΈπΈ =12ππ
ππ2 +12ππππ2π₯π₯2 + πππππ₯π₯
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Toward quantization
category of classical physical quantities
πΏπΏ
ππ = ππβ¨πΏπΏβ¨ππβ1πΏπΏβ¨πΏπΏ
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category of quantum observables
β β β β β πΏπΏ βππ
ππβ¨πΏπΏ
πΏπΏβ = πΏπΏβ1ππ
πΈπΈ = ππβ¨πΏπΏ2β¨ππβ2
1
π₯π₯ + π¦π¦
π¦π¦β β πΏπΏ β
β β πΏπΏ β
π₯π₯
Hom π΄π΄ β,π΅π΅ β isrequired to be a vector space.
β β ππ βπππ₯π₯
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Category of quantum observables
object: asterisk marked by classical quantityarrow: quantum observableEach hom-set is a vector space over β.composition: product of quantum observables
β β πΏπΏ βπ₯π₯
πππ₯π₯ππβ¨πΏπΏ ββ βπππ₯π₯ β π₯π₯
π₯π₯
β β πΏπΏ βπ₯π₯
πππ₯π₯ππ β β ππβ¨πΏπΏ β
ββπππ₯π₯
πππ₯π₯ β π₯π₯ β π₯π₯ β πππ₯π₯26
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Algebraic equation of quantum observables
Description of harmonic oscillator hamiltonian
β β πΏπΏ ββ πΏπΏ2 β β ππβ¨πΏπΏ2β¨ππβ2 βπ₯π₯
πππ₯π₯β
π₯π₯12ππππ
2
ππ β β ππ2 β β ππβ1β¨ππ2 βπππ₯π₯ 1
2ππ
π»π» =12ππ
πππ₯π₯2 +12ππππ2π₯π₯2
β ππβ¨πΏπΏ2β¨ππβ2 β27
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Representation functor sends objects to Hilbert spaces and observables to operators
Category of Hilbert spaces with physical-quantity coefficients
οΏ½π»π» =12ππ
οΏ½ΜοΏ½ππ₯π₯2 +12ππππ2 οΏ½π₯π₯2
β ππβ¨πΏπΏ2β¨ππβ2β¨β
β β πΏπΏβ¨β β πΏπΏ2β¨β β ππβ¨πΏπΏ2β¨ππβ2β¨βοΏ½π₯π₯
οΏ½πππ₯π₯ β
οΏ½π₯π₯12ππππ
2οΏ½1
ππβ¨β β ππ2β¨β β ππβ1β¨ππ2β¨β12πποΏ½1οΏ½πππ₯π₯
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Eigenvector subspace is an equalizer
Category of Hilbert spaces with physical-quantity coefficients.Eigenvalue problem: οΏ½π»π»| β©ππ = ππ| β©ππ
οΏ½π»π»β β πΈπΈβ¨β
πποΏ½1
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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:
β β πΏπΏ β ππ β β πΏπΏβ¨ππ β7. Hom-set is a vector space over β.8. This structure guarantees
homogeneity of physical dimensions of terms in equations of observables.
π₯π₯ π₯π₯
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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:
β β πΏπΏβ¨β ππβ β πΏπΏβ¨ππβ¨β11. Spectral values of observable
operators have suitable physical dimensions.
οΏ½π₯π₯ οΏ½π₯π₯
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References
1. Tanimura, βTopology, category, and differential geometry: from the viewpoint of dualityβ (in Japanese)
2. Tanimura, Series of lectures on geometry and physics published in Mathematical Science (in Japanese)
3. Kitano, βMathematical structure of unit systemsβ J. Math. Phys. 54, 052901 (2013)
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Thank you for your attention
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