(Fundamental) Physics of Elementary Particles Relativistic kinematics (more detail), particle decays & sca t ering, etc. Tristan Hübsch Department of Physics and Astronomy Howard University, Washington DC Prirodno-Matemati č ki Fakultet Univerzitet u Novom Sadu Tuesday, November 1, 11
Fundamental Physics of Elementary Particles Program Relativistic Kinematics (More Detail) Space & time mixing, “usual” relativistic e ff ects, revisited De fi nitions and some crucial minuses Particle Decay 2-particle decay kinematics n -particle decay generalization Particle Collisions & Sca t ering T e CM-system vs . the target system Fusing collisions, and process threshold Invariance & covariance vs . conservation Quantum Kinematics — T e Heisenberg Zone Charge Conservation — Gauge & Other Charges 2 Tuesday, November 1, 11
Relativistic Kinematics Lorentz Transformations T e space-time coordinate transformations that leave Maxwell’s equations invariant mix space and time: r 0 = ~ r 0 + ( γ � 1 )( ˆ r 0 ) ˆ vt 0 , r + ( γ � 1 )( ˆ r ) ˆ r = ~ v + γ ~ ~ v · ~ v � γ ~ ~ v · ~ vt , r 0 t � ~ v · ~ t 0 + ~ v · ~ r t 0 = γ ⇣ ⌘ ⇣ ⌘ t = γ , , c 2 c 2 ⌘ � 1 v 2 1 � ~ ~ 2 , v ⇣ γ : = v : = ˆ v 2 . p c 2 ~ Or, put another way: r 0 = ~ r 0 r 0 ? + γ ( r k � ~ v t ) , r = ~ ? + γ ( k + ~ v t ) ~ ~ ~ ~ r …positions change in the direction of motion only. 3 Tuesday, November 1, 11
Relativistic Kinematics Consequences of Lorentz Transformations Relativity of simultaneity ( t A = t B ): v · ( A ) t i � ~ v · ~ B = γ ~ ~ r B � ~ r i r ⇣ ⌘ t 0 t 0 A � t 0 i = γ i = A , B , , ) c 2 c 2 FitzGerald-Lorentz Relativity of length/distance/extent: contraction r 0 4 ~ k = γ 4 ~ r 0 = 4 r k , 4 ~ r + ( γ � 1 )( ˆ v · 4 r ) ˆ v = 4 r ? + γ 4 ~ ~ ~ ~ r k , r 0 4 ~ ? = 4 ~ r ? , Relativity of duration/passage of time: time dilation r 0 r 0 v · ( ~ A ) A ) + γ ~ B � ~ 4 t = γ 4 t 0 , t B � t A = γ ( t 0 B � t 0 . c 2 Relativity of …well, relative velocities: u 0 u 0 k + ~ ~ u : = 4 v ~ ~ r ? u 0 u 0 u 0 4 t = � + k = ( ~ v ) ˆ v = 0. ~ ~ ~ � , k · ˆ v , ? · ˆ u 0 ) u 0 ) 1 + ( ~ v · ~ 1 + ( ~ v · ~ � � γ c 2 c 2 addition of velocities 4 Tuesday, November 1, 11
Relativistic Kinematics Conventions and Details Frequently useful expansions: 1 − b 2 ≈ 1 + 1 1 2 b 2 + 3 8 b 4 + 5 16 b 6 + O b : = v 2 b 8 � � g = , c 2 ⌧ 1; p 1 � | ~ v | ⇣ ⌘ e : = ⌧ 1. 1 1 + 1 4 e + 3 5 h e 4 �i 32 e 2 + 128 e 3 + O c � and . √ ≈ 2 e Notation (Cartesian coord’s): 3 3 x µ ˆ x 0 = ct , ~ x i ˆ ∑ ∑ x : = where e µ , r = e i , µ = 0 i = 1 …so Lorentz transformations are linear: L 00 L 01 L 02 L 03 x 0 y 0 L 10 L 11 L 12 L 13 x 1 y 1 . y µ = L µ ν x ν , L y = L L L x = ⇔ ⇔ y 2 L 20 L 21 L 22 L 23 x 2 y 3 L 30 L 31 L 32 L 33 x 3 5 Tuesday, November 1, 11
Relativistic Kinematics Conventions and Details Lorentz boosts: − γ v y − γ v x − γ v z 2 3 γ c c c 1 + ( γ − 1 ) v 2 ( γ − 1 ) v x v y − γ v x ( γ − 1 ) v x v z x 6 7 v 2 v 2 v 2 c ~ ~ ~ 6 7 L = L L L v 2 6 7 − γ v y ( γ − 1 ) v y v x ( γ − 1 ) v y v z y 1 + ( γ − 1 ) 6 7 v 2 v 2 v 2 c ~ ~ ~ 4 5 1 + ( γ − 1 ) v 2 − γ v z ( γ − 1 ) v z v x ( γ − 1 ) v z v z z c v 2 v 2 v 2 ~ ~ ~ ∂ L µ ν T η L − 1 , ∂ x ρ = 0, ( µ , ν , ρ = 0, 1, 2, 3 ) , η = η det ( L L ) = 1, L η η L L L L L η η η L L L leave invariant the “proper time,” τ : c 2 τ 2 = x 2 = x · x : = x µ η µ ν x ν , where [ η μν ] = diag[1,–1,–1,–1] is the spacetime metric. s = – c τ is the “interval,” d s 2 = – c 2 d τ 2 the “line element.” 6 Tuesday, November 1, 11
Relativistic Kinematics Conventions and Details T e o f -cited special case: 2 3 � γ v 0 0 γ 8 = : c tanh ( φ ) , c v � γ v < γ 0 0 6 7 = cosh ( φ ) , c [ L µ ν ] = γ 6 7 1 0 0 0 1 0 4 5 : v = sinh ( φ ) ; c γ 0 1 0 0 0 1 is related (by analytic continuation) to a rotation: 2 3 2 3 2 3 c ( it 0 ) cos ( φ ) � sin ( φ ) c ( it ) 0 0 x 0 1 x 1 sin ( φ ) cos ( φ ) 0 0 6 7 6 7 6 7 5 = 6 7 6 7 6 7 x 0 2 x 2 0 0 1 0 4 4 5 4 5 x 0 3 x 3 0 0 0 1 and the coordinates ( x 0 = cit , x 1 , x 2 , x 3 ) span the “World of (Hermann) Minkowski;” but t → it is “Wick-rotation.” 7 Tuesday, November 1, 11
Relativistic Kinematics Energy and Momentum (Hamiltonian action) ∝ (“world line” length): Z B Z t B d t α c A d ( c τ ) α ( 1.8 ) S = − = − γ , t A r 2 α c v 2 ⇣ v 4 1 − v 2 c 2 ≈ − α c + 1 ⌘ L = − α c c 2 + α c O c 4 the non-relativistic expansion (1.10c). Since the initial …where we set α = mc , leading to: r r v 2 1 − ~ 1 − 1 c 2 | . L = − mc 2 γ − 1 = − mc 2 = − mc 2 r | 2 . ~ c 2 T ereupon: p : = ∂ L = ∂ L ( v = m γ ~ ~ v ∂ . ∂ ~ ~ r p · . v + mc 2 γ − 1 = m γ c 2 , E : = ~ ~ r − L = m γ ~ v · ~ 8 h Tuesday, November 1, 11
Relativistic Kinematics Energy-Momentum Note: E = γ mc 2 is total energy ; E 0 = mc 2 is rest energy . Also: T = E – E 0 = m ( γ –1) c 2 is kinetic energy . T e energy-momentum 4-vector (4-momentum) is p = ( p µ ) : = ( − E / c , ~ p ) = ( − m γ c , m γ ~ v ) . the Lorentz-invariant square of which is: p 2 : = p µ η µ ν p ν = E 2 / c 2 − ~ p 2 = m 2 γ 2 c 2 − ~ p 2 1 − v 2 2 = m 2 γ 2 c 2 ⇣ ⌘ = m 2 c 2 . c 2 T is de fi nes the Lorentz-invariant mass: ( just like c 2 τ 2 = x · x = c 2 t 2 − ~ m 2 c 4 = p · p = E 2 − ~ p 2 c 2 r 2 ) and the Lorentz-invariant combination of E , p x , p y & p z . 9 Tuesday, November 1, 11
Relativistic Kinematics Conventions and Details Now about that sign in : p = ( � E / c , ~ p ) Starting from quantum theory in coord. representation, mechanics, where in coordinate h ∂ become p µ = ¯ ∂ x µ : i we identify: h ∂ h ∂ h ∂ h p 0 = ¯ ∂ x 0 = ¯ ∂ ( ct ) = � 1 ∂ t = � 1 p = + ¯ ~ ~ c i ¯ r . c H , i i i T e same may be derived by purely classical arguments: s Z t B Z x 0 v 2 1 − ~ ( v µ ) : = ∂ x µ ∂ t = ( c , . x 1 , . x 2 , . B d x 0 L 0 , d t mc 2 x 3 ) . S = − c 2 = − x 0 t A A √ � v 2 � c 2 − ~ ∂ − m ( p 0 p v 2 , c 2 − ~ : = c = − m γ c = − E / c , L 0 : = m ∂ c √ � v 2 � ∂ L 0 = ∂ L 0 x µ = c ∂ L 0 c 2 − ~ ∂ − m = m γ δ ij v j , p i : = c p µ : = ∂ v µ , c ∂ . ∂ v i ∂ ∂ x µ 1 ∂ x 0 10 Tuesday, November 1, 11
Decays and Collisions General Remarks Strictly conserved quantities the sum of (observable) 4-momenta the sum of (observable) angular momenta (incl. spin) the sum of (observable) Noether charges (incl. EM ch.) Collisions can be: Type Kinetic Energy Mass Elastic Conserved Conserved Fissile/Explosive increased decreased Fusing/Implosive decreased increased 11 Tuesday, November 1, 11
Particle Decays 2-Particle Decay Consider A → B + C , with m A ≠ 0. ! p A = ( � m A c , ~ Use the A -rest frame: , and 0 ) , p B = ( − E B / c , ~ p B ) , p C = ( − E C / c , ~ p C ) , 4-momentum conservation: p A = p B + p C , implies that − m A c = − E B / c − E C / c ~ p B = − ~ and p C T is is useful, but provides no relationship between the energies and the 3-momenta. So, consider p A = p B + p C also as a 4- vector equation. T is equation is (by de fi nition) not invariant, but p A 2 =(p B +p C ) 2 , p B 2 =(p A –p C ) 2 and p C 2 =(p A –p B ) 2 are. 12 Tuesday, November 1, 11
Particle Decays 2-Particle Decay So, consider p A 2 =(p B +p C ) 2 : A = ( p B + p ) 2 = p 2 p 2 B + p 2 C + 2 p B · p C , · k k k k ⇣ E B E C ⇣ ⌘ B c 2 + m 2 C c 2 + 2 m 2 m 2 A c 2 c � ~ c � p B · ~ p C , · c c k C c 2 + 2 E B E C B c 2 + m 2 m 2 p B 2 . + 2 ~ c 2 …so there is a relationship between energies and 3- momenta (and masses)! But, it’s complicated. 13 Tuesday, November 1, 11
Particle Decays 2-Particle Decay Consider instead p B 2 =(p A –p C ) 2 : B = ( p A � p C ) 2 = p 2 p 2 A + p 2 C � 2 p A · p C , k k C c 2 � 2 E A E C A c 2 + m 2 m 2 m 2 B c 2 c , c k A c 2 + m 2 C c 2 � 2 m A E C . m 2 T is is immediately solved: And p C 2 =(p A –p B ) 2 similarly yields the ⇣ m 2 A + m 2 C − m 2 ⌘ c 2 . B B ↔ C result for E B . E C = 2 m A 14 Tuesday, November 1, 11
Particle Decays 2-Particle Decay q Use the universal ( on-shell ) relativistic relationship: s s⇣ m 2 E 2 A + m 2 C − m 2 ⌘ 2 C c 2 = c c 2 − m 2 B − m 2 C | ~ p C | = C , 2 m A p ( m A + m B + m C )( m A − m B + m C )( m A + m B − m C )( m A − m B − m C ) = c , 2 m A q m 4 A + m 4 B + m 4 C − 2 m 2 A m 2 B − 2 m 2 A m 2 C − 2 m 2 B m 2 C = c E C I T 2 m A O N e r a e s e h t …and recall: f o l l A ! s t n a t s n o c ~ p B = − ~ p C ⇣ m A 2 + m B 2 − m C 2 ⇣ m A 2 + m C 2 − m B 2 ⌘ ⌘ c 2 , E C = c 2 . E B = 2 m A 2 m A 15 Tuesday, November 1, 11
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