Algebra in Flatland Connie Dennis*, Cassie Stamper* Department of Mathematics Kansas State University Manhattan, KS 66506 Mentor: Dr. David Yetter* July 24, 2012 * Supported by NSF grant # DMS1004336
Motivation C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 2 / 35
Motivation The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 2 / 35
Motivation The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 2 / 35
Motivation The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. This is the idea of quantum computing. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 2 / 35
Motivation The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. This is the idea of quantum computing. Physics in two spatial dimensions is different than physics in three and higher dimensions. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 2 / 35
Motivation The starting point is to think of computations as physical things that take place in space and time, rather than abstractions. Secondly, we wanted to compute things by representing mathematical quantatities by quantum mechanical states. This is the idea of quantum computing. Physics in two spatial dimensions is different than physics in three and higher dimensions. In three and higher dimensions there are only two types of particles: bosons and fermions. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 2 / 35
Motivation Continued C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 3 / 35
Motivation Continued If two bosons in identical states are swapped, the state remains the same. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 3 / 35
Motivation Continued If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 3 / 35
Motivation Continued If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. In a plane, quasi particles can exhibit the fractional quantum hall effect. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 3 / 35
Motivation Continued If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. In a plane, quasi particles can exhibit the fractional quantum hall effect. When two quasi particles are swapped, the state is multiplied by a complex number with absolute value 1. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 3 / 35
Motivation Continued If two bosons in identical states are swapped, the state remains the same. If two fermions in identical states are swapped, the state negates. In a plane, quasi particles can exhibit the fractional quantum hall effect. When two quasi particles are swapped, the state is multiplied by a complex number with absolute value 1. The motivation for this project is to represent math in quantum mechanical states, where when two things are swapped, the state is multiplied by a phase ζ which is a primitive N th root of unity. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 3 / 35
Need to Know Definition Let Z be the set of all integers. A Z graded vector space is a vector space, V which decomposes into a direct sum of the form: � V = V n n ∈ Z Where each V n is a vector space. For a given n the elements of V n are then called homogeneous elements of degree n. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 4 / 35
Need to Know Continued We will use || to represent degree. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 5 / 35
Need to Know Continued We will use || to represent degree. The anyonic braiding associated to ζ ( ζ N = 1 primitive ) is given by: ∀ a,b homogeneous σ ( a ⊗ b ) = ζ | a || b | b ⊗ a C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 5 / 35
Questions Question What happens to ordinary commutativity and associativity in the anyonic setting? C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 6 / 35
Questions Question What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ | a || b | b · a C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 6 / 35
Questions Question What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ | a || b | b · a Answer 2: Associative stays the same unless the multiplication changes degrees. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 6 / 35
Questions Question What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ | a || b | b · a Answer 2: Associative stays the same unless the multiplication changes degrees. Answer 3: If | m | � = 0 we will answer later. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 6 / 35
Questions Question What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ | a || b | b · a Answer 2: Associative stays the same unless the multiplication changes degrees. Answer 3: If | m | � = 0 we will answer later. Question Do the resulting axioms give reasonable algebraic systems? C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 6 / 35
Questions Question What happens to ordinary commutativity and associativity in the anyonic setting? Answer 1: Commutative became a · b = ζ | a || b | b · a Answer 2: Associative stays the same unless the multiplication changes degrees. Answer 3: If | m | � = 0 we will answer later. Question Do the resulting axioms give reasonable algebraic systems? We used rewrite systems as a tool to answer this question. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 6 / 35
Definitions C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 7 / 35
Definitions Definition A rewrite system is a collection of directed rules for replacing parts of symbol strings with other symbol strings. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 7 / 35
Definitions Definition A rewrite system is a collection of directed rules for replacing parts of symbol strings with other symbol strings. Definition Descending Chain Condition- given a word there are no infinite strings of successive rule applications which can be made starting at the word. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 7 / 35
Definitions Definition A rewrite system is a collection of directed rules for replacing parts of symbol strings with other symbol strings. Definition Descending Chain Condition- given a word there are no infinite strings of successive rule applications which can be made starting at the word. Definition Local Confluence- If two instances apply to a word, then there are sequences of rule applications to each of the results which give equal results. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 7 / 35
Knuth - Bendix Theorem Knuth - Bendix Theorem: If a rewrite system satisfies two properties, the Descending Chain Condition (DCC) and Local Confluence, then: (a) Given any word, there is a unique reduced word. The reduced word is the canonical representative of the equivalence class. (b) If two words are equivalent, the reduced word reachable from each is the same. C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 8 / 35
Free Anyonic Commutative Algebra C [ x ( d 1 ) .... x ( d n ) ] ζ n 1 C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 9 / 35
Free Anyonic Commutative Algebra C [ x ( d 1 ) .... x ( d n ) ] ζ n 1 freely generated by x 1 ...... x n C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 9 / 35
Free Anyonic Commutative Algebra C [ x ( d 1 ) .... x ( d n ) ] ζ n 1 freely generated by x 1 ...... x n of degrees d 1 ...... d n C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 9 / 35
Free Anyonic Commutative Algebra C [ x ( d 1 ) .... x ( d n ) ] ζ n 1 freely generated by x 1 ...... x n of degrees d 1 ...... d n ζ N = 1 C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 9 / 35
Free Anyonic Commutative Algebra C [ x ( d 1 ) .... x ( d n ) ] ζ n 1 freely generated by x 1 ...... x n of degrees d 1 ...... d n ζ N = 1 ∀ a , b homogenous : C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 9 / 35
Free Anyonic Commutative Algebra C [ x ( d 1 ) .... x ( d n ) ] ζ n 1 freely generated by x 1 ...... x n of degrees d 1 ...... d n ζ N = 1 ∀ a , b homogenous : a · b = ζ | a || b | b · a C. Dennis, C.Stamper (KSU) SUMaR 2012 July 24, 2012 9 / 35
Recommend
More recommend
