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IllustraKon of Category Hilbwith examples in
Atomic and OpKcal Physics M.I.T. Student
Department of PhysicsMIT
May 13, 2013
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Outline
• What is Atomic and OpKcal physics?e.g. atoms and photons
• What is Hilbert space?• What is Category Hilb?• Monoidal structure of Hilb• Feynman diagram in the language of category
e.g. Quantum Harmonic oscillator• Conclusions and interests
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What is Atomic and OpKcal Physics about?
Physics: Quantum phenomena of parKcles*Atoms, from one to an ensemble*Photons, from one to a bunch, from lasers, or trapped in a cavity
*simulate other quantum systemsuse opKcal laNce to simulate crystaluse quantum double well to simulate Josephson juncKon
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What is Atomic and OpKcal Physics about?
Math: Hilbert spaceDef. A real or complex inner product space that is also a complete metricspace with respect to the distance funcKon induced by the inner product.
* One atom in a harmonic trap induced by lasers
2p 1 1ˆ m!2 H = + x 2 = ( + n)~!2m 2 En 2X
x = {X
ci|�i > |ci 2 C, |ci|2= 1}i i
* A cavity that trapped any number of photons
H =1 ✏E2 Ej = j~!
2 Xy = {
X cj |j > |cj 2 C, |cj |2= 1}
j j
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What is Atomic and OpKcal Physics about?
* a single atom with two hyperfine states
|2z = {cs|s > +cp|p > ||cs +|cp|2= 1}
* a single atom with two hyperfine statesinteracKng with photons in a cavity
w = z ⇥ y
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Category Hilb
Content removed due to copyright restrictions.
To view definitions of category Hilb, go to: "Hilb category of Hilbert Spaces" by bc1. http://planetmath.org/sites/default/files/texpdf/41070.pdf
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Category Hilb
• Ob(Hilb) X
x = {X
ci|ri > |ci 2 C, |ci|2= 1} i i X
y = {X
cj |j > |cj 2 C, |cj |2= 1}j j
|2z = {cs|s > +cp|p > ||cs +|cp|2= 1}
w • Morphism
Hom_Hilb (x, x) : ˆ p, ˆ p... x, ˆ x + ˆHom_Hilb (y, y) : E, e↵E
2 ...
Hom_Hilb (x, y) :m : |¢i >! |i >, i = 0, 1, 2...
Physics: phonon – photon mapping, massless, parKcle – wave duality
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|s>
|0, p >
Category Hilb is monoidal
• Physics: InteracKon of two system -‐> joint system Math: tensor product -‐> monoidal category
⇤(y, z) = w• Monoidal category[2] i) a category Hilb ii) a functor
|1> ⇤ : Hilb ⇥ Hilb ! Hilb Physics: morphism is as important asobjects, so that we can form �� ⇥ ˆ ( ˆ a†)(|p >, |0 >) = |s, 1 > interacKon in the Hamiltonian
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Category Hilb is monoidal
iii) a unit objectC 2 Hilb
iv) le< unit law: la : C ⇥ a ! a right unit law: ra : a ⇥ C ! a Physics: states in Hilbert space is normalized.
natural isomorphism called the associator: Content removed due to copyright restrictions.
To view the diagram, go to: "Section 4. The Monoidal Category of Hilbert Spaces" by John Baez.http://math.ucr.edu/home/baez/quantum/node4.html
such that the following diagrams commute for all objects A, B, C, D,v)
Content removed due to copyright restrictions.
To view the diagram, go to: "Section 4. The Monoidal Category of Hilbert Spaces" by John Baez.http://math.ucr.edu/home/baez/quantum/node4.html
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Category Hilb is monoidal
• Associator is natural in a precise sense Content removed due to copyright restrictions. To view the diagram, go to: "Section 4. The Monoidal Category of Hilbert Spaces" by John Baez.http://math.ucr.edu/home/baez/quantum/node4.html
Physics:S, T, L can be viewed as base transformaKon. The diagram is commute indicates that associator is defined in a base-‐
independent manner.
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Feynman diagram
• states processed byevoluKon operators Exp(-‐iHt) interacKon operators o+ˆ (oAa† + ˆ a) evoluKon operators to final states
• This is true for all Hilbert spaces.• For one Hilbert space that has a specific interacKon form, one can also
categorize it alone.
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Feynman diagram[3] • Quantum harmonic oscillator represented in Fock space.
Categorize Fock space to FinSet_0Categorize states to a functors from a Grp to FinSet_0
Inner Product: a Grp
Operators: a Grp with two “projecKon” functorsinto FinSet_0.
Diagrams are taken from the paper "Categorifying the Quantum Harmonic Oscillator" by Jeffrey Morton and are used by permission.
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Feynman diagram
Kme evoluKon phonon creaKon and annihilaKon
Diagrams are taken from the paper "Categorifying the Quantum Harmonic Oscillator" by Jeffrey Morton and are used by permission.
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Feynman diagram
a†a†aa : atom-‐atom short range interacKon in our lab
Diagrams are taken from the paper "Categorifying the Quantum Harmonic Oscillator" by Jeffrey Morton and are used by permission.
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Conclusions and interests
• Hilb has Hilbert space as objects, operators as morphisms• Hilb is a monoidal category
Quantum systems can have interacKon with each other, joining togetheras a new system• More than one way to categorize quantum systems
Categorize Fock space into FinSet_0, due to observaKon of its relaKonships with enumeraKve combinatorics. • Feynman diagram can be translated to diagrams in category.
* Non-‐trivial morphism from finite Hilbert space to an infinite one or viceversa?* Benefit of Feynman diagram in category language?* Similarity of nCob and Hilb[4], relaKon of general relaKvity and quantummechanics? Physics predicaKon? * Topological field theory, quantum gravity…
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Bibliography
[1] bci1, Hilb category of Hilbert spaces hMp://planetmath.org/sites/default/files/texpdf/41070.pdf[2] John Baez, The Monoidal Category of Hilbert Spaces, 2004http://math.ucr.edu/home/baez/quantum/node4.html[3] Jeffrey Morton, Categorifying the Quantum Harmonic Oscillator, 2006 hMp://math.ucr.edu/home/baez/morton_ct06talk.ps [4] John Baez, Quantum Quandaries: A Category -‐TheoreKc PerspecKve, 2004http://arxiv.org/abs/quant-ph/0404040v2 [5] David Spivak, Category Theory for ScienKsts, 2013hMp://arxiv.org/pdf/1302.6946.pdf [6] Bob Coecke, Kindergarten Quantum Mechanics, 2005 http://arxiv.org/abs/quant-ph/0510032 [7] John Baez, James Dolan, From Finite Sets to Feynman Diagrams, 2000 hMp://arxiv.org/abs/math/0004133
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Bibliography
[8] John Baez, Mike Stay, Physics, Topology, Logic and ComputaKon: ARoseMa Stone, 2009hMp://arxiv.org/pdf/0903.0340v3.pdf [9] Chris Heunen, Categorical quantum models and logics, 2010hMp://books.google.com/books? id=nsd8EExdKIwC&printsec=frontcover#v=onepage&q&f=false [10] nLabhMp://ncatlab.org/nlab/show/HomePage
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Thank you!
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