Magnetic monopoles in noncommutative quantum mechanics Samuel Kov - PowerPoint PPT Presentation

Magnetic monopoles in noncommutative quantum mechanics Samuel Kov´ aˇ cik Commenius University Bratislava (soon) Dublin Institute for Advanced Studies arXiv:1604.05968 25.8.2016 Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 1 / 7

Magnetic monopoles Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 2 / 7

Magnetic monopoles # of observed magnetic monopoles ≈ # of observed unicorns Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 2 / 7

Magnetic monopoles # of observed magnetic monopoles < # of observed unicorns Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 2 / 7

Magnetic monopoles Figure: Elasmotherium sibiricum , Giant Siberian Unicorn, extinct Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 2 / 7

Magnetic monopoles Maxwell’s equations div � E ( � r , t ) = 4 πρ E ( � r , t ) , div � B ( � 4 πρ M ( � r , t ) = r , t ) , − ∂ � B ( � r , t ) rot � − 4 π � E ( � r , t ) = J M ( � r , t ) , ∂ t ∂ � E ( � r , t ) rot � + 4 π � B ( � r , t ) = J E ( � r , t ) . ∂ t Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 3 / 7

Magnetic monopoles Maxwell’s equations div � E ( � r , t ) = 4 πρ E ( � r , t ) , div � B ( � r , t ) = 4 πρ M ( � r , t ) , − ∂ � B ( � r , t ) rot � − 4 π � E ( � r , t ) = J M ( � r , t ) , ∂ t ∂ � E ( � r , t ) rot � + 4 π � B ( � r , t ) = J E ( � r , t ) . ∂ t Ordinary QM v − µ� r � = � r × � r , L DQC: µ = eg ∈ Z / 2 , i µε ijk � r [ π i , π j ] = r 3 , π are canonical momenta, Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 3 / 7

Magnetic monopoles One of the safest bets that one can make about physics not yet seen, Polchinski 2012 Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 3 / 7

NC space Three dimensional rotationally invariant NC space defined by λ ≈ l Planck. R 3 λ x i = λσ i αβ a + α a β , r = λ ( a + α a α + 1); α, β = 1 , 2 where σ i are the Pauli matrices and the c/a operators satisfy 1 ) n 1 ( a + β ] = 0 , | n 1 , n 2 � = ( a + 2 ) n 2 [ a α , a + β ] = δ αβ , [ a α , a β ] = [ a + α , a + √ n 1 ! n 2 ! | 0 � Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 4 / 7

Physical states Hilbert space H λ Spanned on analytic functions equipped with a norm 2 , � Ψ � 2 = 4 π λ 2 Tr[ˆ r Ψ † Ψ] . � C ( m , n ) a + m 1 a + m 2 a n 1 1 a n 2 Ψ = 1 2 Special attention is paid to Ψ κ jm κ = m 1 + m 2 − n 1 − n 2 . Operators in H ˆ 2 m λ r [ a + 1 H 0 Ψ = α , [ a α , Ψ]] L j ] = i ε ijk ˆ ˆ 2 λ [ x i , Ψ] , [ˆ L i , ˆ 1 L i Ψ = L k ˆ X i Ψ = 1 r Ψ = 1 2 ( x i Ψ + Ψ x i ) , ˆ 2 ( r Ψ + Ψ r ) Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 5 / 7

Monopole states ↔ generalized states Ψ κ ( e − i τ a + , e i τ a ) = e − i τκ Ψ κ ( a + , a ) , τ ∈ R , fixed κ ∈ Z , Comparison MM vs κ states [ ˆ X i , ˆ X j ] = λ 2 ε ijk ˆ [ˆ x i , ˆ x j ] = 0 ↔ L k , � 1 − λ 2 ˆ � [ ˆ X i , ˆ [ˆ x i , ˆ π j ] = i δ ij ↔ V j ] = i δ ij H 0 , ˆ x k ˆ = i − κ X k � � V i , ˆ ˆ [ˆ π i , ˆ π j ] = i µε ijk ↔ 2 ε ijk r 2 − λ 2 ) . V j r 3 ˆ r (ˆ C 1 = κ ˆ ˆ C 1 = − q µ ↔ 2 q , � κ � 2 C 2 = q 2 + ( µ ) 2 ( − 2 E ) C 2 = q 2 + ˆ ˆ ( − 2 E + λ 2 E 2 ) . ↔ 2 Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 6 / 7

Monopole states ↔ generalized states Ψ κ ( e − i τ a + , e i τ a ) = e − i τκ Ψ κ ( a + , a ) , τ ∈ R , fixed κ ∈ Z , Comparison MM vs κ states [ ˆ X i , ˆ X j ] = λ 2 ε ijk ˆ [ˆ x i , ˆ x j ] = 0 ↔ L k , � 1 − λ 2 ˆ � [ ˆ X i , ˆ [ˆ x i , ˆ π j ] = i δ ij ↔ V j ] = i δ ij , H 0 ˆ x k ˆ = i − κ X k � � V i , ˆ ˆ [ˆ π i , ˆ π j ] = i µε ijk ↔ 2 ε ijk r 2 − λ 2 ) . V j r 3 ˆ r (ˆ C 1 = κ ˆ ˆ C 1 = − q µ ↔ 2 q , � κ � 2 C 2 = q 2 + ( µ ) 2 ( − 2 E ) C 2 = q 2 + ˆ ˆ ( − 2 E + λ 2 E 2 ) . ↔ 2 µ ∈ Z / 2 ↔ − κ/ 2 ∈ Z / 2 . Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 6 / 7

Thank you for your attention. Samuel Kov´ aˇ cik (KTF FMFI) NC QM 25.8.2016 7 / 7