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Lecture 26. Introduction to Relativistic Classical and Quantum Mechanics
(Ch. 2-5 of Unit 2 4.03.12)
Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
(Includes Lecture 25 review)
1Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
2Tuesday, April 3, 2012
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Introducing the “Sin-Tan Rosetta Stone”
https://www.uark.edu/ua/pirelli/php/hyper_constrct.php
NOTE: Angle φ is now called stellar aberration angle σ
Fig. C.2-3and
Fig. 5.4in Unit 2
3Tuesday, April 3, 2012
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More about the“Sin-Tan Rosetta”
Note identities
4Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsGeometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
5Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts
(From Fig. 2.3.4)
GGrroouupp vveelloocciittyy u aanndd pphhaassee vveelloocciittyy c2/uaarree hhyyppeerrbboolliicc ttaannggeenntt ssllooppeess
Newtonian speed u~cρRelativisticgroup wavespeed u=c tanh ρ
LLooww ssppeeeedd aapppprrooxxiimmaattiioonn
33−
== BB==22ϖ
433221100--11
4
--22
BB ccoosshh ρ
BB ssiinnhh ρBB ee--ρ ck
ω
== 22BB==442ϖ
Δω
c Δk
PPGG
BB ee++ρ
Ghyperbolas
P hyperbolasc line
cc
ω
ck
u
cu
c dω =dcckkcckkω=
Group velocity
k=BB ssiinnhh ρω=BB ccoosshh ρ
ω =cckkccuu
Phase velocity
== BBϖ
Rare but important case where
with LARGE Δk(not infinitesimal)
dωdk =
ΔωΔk
PPΔω
c Δk
ccuu
BB ssiinnhh ρ
BB ccoosshh ρ
Rapidity ρ approaches u/c
Lecture 25 ended here 6Tuesday, April 3, 2012
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PPΔω
c Δk
BB ssiinnhh ρ
BB ccoosshh ρ
Group vs. phase velocity and tangent contacts
7Tuesday, April 3, 2012
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PPΔω
c Δk
BB ssiinnhh ρ
BB ccoosshh ρ
8Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
9Tuesday, April 3, 2012
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Start with low speed approximations : ω = Bcoshρ = B(1+2
1 ρ2 + ...) where: ρ uc
10Tuesday, April 3, 2012
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Start with low speed approximations : ω = Bcoshρ = B(1+2
1 ρ2 + ...) where: ρ uc
11Tuesday, April 3, 2012
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Start with low speed approximations : ω = Bcoshρ = B(1+2
1 ρ2 + ...) where: ρ uc
12Tuesday, April 3, 2012
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Start with low speed approximations : ω = Bcoshρ = B(1+2
1 ρ2 + ...) where: ρ uc
13Tuesday, April 3, 2012
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14Tuesday, April 3, 2012
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15Tuesday, April 3, 2012
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cp'=hck'cctt
cctt''
EnergyE=hω
Momentumcp=hck
Mc2
ωm=49ω1
76543210-1-2-3-4-4-6m
36
251694
(a) Einstein-Planck Dispersion
(b) DeBroglie-Bohr Dispersion
E'=hω'E2 - c2p2 =(Mc2)2
photon: M=0E = c p
E = p2/2M
E = B m2
tachyon:
Bohr - Schrodinger Dispersion
Einstein - Planck Dispersion
16Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
17Tuesday, April 3, 2012
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18Tuesday, April 3, 2012
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19Tuesday, April 3, 2012
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20Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
21Tuesday, April 3, 2012
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Mrest(γ)=0 Mmom(γ)=p/c=k/c=ω/c2 Meff(γ)=∞
Equations (4.11) in Unit 2
Three kinds of mass for photon γ in CW relativistic theory What’s the matter with light?
(1)Einstein rest mass (2) Galilean momentum mass (3) Newtonian inertial mass Mrest= Mmom=p/u= Mmom=F/a=
ω proper
c2
kdωdk
d 2ωdk 2
A 2-CW 600THz cavity has zero total momentum p, but each photon adds a tiny mass Mγ to it.
Mγ=ω/c2=ω (1.2·10-51)kg·s= 4.5·10-36kg (for: ω = 2π·600THz )
A 1-CW state has no rest mass, but 1-photon momentum is a non-zero value pγ=Mγ c. (Galilean revemge II.)
pγ=k=ω/c=ω (4.5·10-43)kg·m=1.7·10-27kg·m·s-1 (for: ω = 2π·600THz )
22Tuesday, April 3, 2012
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Mrest(γ)=0 Mmom(γ)=p/c=k/c=ω/c2 Meff(γ)=∞
Equations (4.11) in Unit 2
Three kinds of mass for photon γ in CW relativistic theory What’s the matter with light?
(1)Einstein rest mass (2) Galilean momentum mass (3) Newtonian inertial mass Mrest= Mmom=p/u= Meff=F/a=
ω proper
c2
kdωdk
d 2ωdk 2
A 2-CW 600THz cavity has zero total momentum p, but each photon adds a tiny mass Mγ to it.
Mγ=ω/c2=ω (1.2·10-51)kg·s= 4.5·10-36kg (for: ω = 2π·600THz )
A 1-CW state has no rest mass, but 1-photon momentum is a non-zero value pγ=Mγ c. (Galilean revemge II.)
pγ=k=ω/c=ω (4.5·10-43)kg·m=1.7·10-27kg·m·s-1 (for: ω = 2π·600THz )
23Tuesday, April 3, 2012
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Mrest(γ)=0 Mmom(γ)=p/c=k/c=ω/c2 Meff(γ)=∞
Equations (4.11) in Unit 2
Three kinds of mass for photon γ in CW relativistic theory What’s the matter with light?
(1)Einstein rest mass (2) Galilean momentum mass (3) Newtonian inertial mass Mrest= Mmom=p/u= Meff=F/a=
ω proper
c2
kdωdk
d 2ωdk 2
A 2-CW 600THz cavity has zero total momentum p, but each photon adds a tiny mass Mγ to it.
Mγ=ω/c2=ω (1.2·10-51)kg·s= 4.5·10-36kg (for: ω = 2π·600THz )
A 1-CW state has no rest mass, but 1-photon momentum is a non-zero value pγ=Mγ c. (Galilean revemge II.)
pγ=k=ω/c=ω (4.5·10-43)kg·m=1.7·10-27kg·m·s-1 (for: ω = 2π·600THz )
24Tuesday, April 3, 2012
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Mrest(γ)=0 Mmom(γ)=p/c=k/c=ω/c2 Meff(γ)=∞
Equations (4.11) in Unit 2
Three kinds of mass for photon γ in CW relativistic theory What’s the matter with light?
(1)Einstein rest mass (2) Galilean momentum mass (3) Newtonian inertial mass Mrest= Mmom=p/u= Meff=F/a=
ω proper
c2
kdωdk
d 2ωdk 2
A 2-CW 600THz cavity has zero total momentum p, but each photon adds a tiny mass Mγ to it.
Mγ=ω/c2=ω (1.2·10-51)kg·s= 4.5·10-36kg (for: ω = 2π·600THz )
A 1-CW state has no rest mass, but 1-photon momentum is a non-zero value pγ=Mγ c. (Galilean revemge II.)
pγ=k=ω/c=ω (4.5·10-43)kg·m=1.7·10-27kg·m·s-1 (for: ω = 2π·600THz )
25Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
26Tuesday, April 3, 2012
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Bohr-Schrodinger (BS) approximation throws out Mc2
E =Mc2
1− u2 / c2= Mc2 coshρ = Mc2 1+ sinh2 ρ = Mc2( )2 + cp( )2
E = Mc2( )2 + cp( )2⎡
⎣⎢
⎤
⎦⎥1/2
≈ Mc2 + 12M
p2 BS−approx⎯ →⎯⎯⎯⎯12M
p2
The BS claim: may shift energy origin (E=Mc2, cp=0) to (E=0, cp=0). (Frequency is relative!)
cp'=hck'cctt
cctt''
EnergyE=hω
Momentumcp=hck
Mc2
ωm=49ω1
76543210-1-2-3-4-4-6m
36
251694
(a) Einstein-Planck Dispersion
(b) DeBroglie-Bohr Dispersion
E'=hω'E2 - c2p2 =(Mc2)2
photon: M=0E = c p
E = p2/2M
E = B m2
tachyon:
Bohr - Schrodinger Dispersion
Einstein - Planck Dispersion
=dkdω
kω
Group velocity u=Vgroup is a differential quantity unaffected by origin shift.
Phase velocity =Vphase is greatly reduced by deleting Mc2 from E=ω.
It slows from Vphase=c2/u to a sedate sub-luminal speed of Vgroup/2.
ω BS (k) = k 2
2M gives: v phase=
ω BS
k= k
2M
and: vgroup=dω BS
dk= kM
27Tuesday, April 3, 2012
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Bohr-Schrodinger (BS) approximation throws out Mc2
E =Mc2
1− u2 / c2= Mc2 coshρ = Mc2 1+ sinh2 ρ = Mc2( )2 + cp( )2
E = Mc2( )2 + cp( )2⎡
⎣⎢
⎤
⎦⎥1/2
≈ Mc2 + 12M
p2 BS−approx⎯ →⎯⎯⎯⎯12M
p2
The BS claim: may shift energy origin (E=Mc2, cp=0) to (E=0, cp=0). (Frequency is relative!)
cp'=hck'cctt
cctt''
EnergyE=hω
Momentumcp=hck
Mc2
ωm=49ω1
76543210-1-2-3-4-4-6m
36
251694
(a) Einstein-Planck Dispersion
(b) DeBroglie-Bohr Dispersion
E'=hω'E2 - c2p2 =(Mc2)2
photon: M=0E = c p
E = p2/2M
E = B m2
tachyon:
Bohr - Schrodinger Dispersion
Einstein - Planck Dispersion
=dkdω
kω
Group velocity u=Vgroup is a differential quantity unaffected by origin shift.
Phase velocity =Vphase is greatly reduced by deleting Mc2 from E=ω.
It slows from Vphase=c2/u to a sedate sub-luminal speed of Vgroup/2.
ω BS (k) = k 2
2M gives: v phase=
ω BS
k= k
2M
and: vgroup=dω BS
dk= kM
28Tuesday, April 3, 2012
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=dkdω
kω
Group velocity u=Vgroup is a differential quantity unaffected by origin shift.
But, Phase velocity =Vphase is greatly reduced by deleting Mc2 from E=ω.
It slows from Vphase=c2/u to a sedate sub-luminal speed of Vgroup/2.
Bohr-Schrodinger (BS) approximation throws out Mc2
E =Mc2
1− u2 / c2= Mc2 coshρ = Mc2 1+ sinh2 ρ = Mc2( )2 + cp( )2
E = Mc2( )2 + cp( )2⎡
⎣⎢
⎤
⎦⎥1/2
≈ Mc2 + 12M
p2 BS−approx⎯ →⎯⎯⎯⎯12M
p2
The BS claim: may shift energy origin (E=Mc2, cp=0) to (E=0, cp=0). (Frequency is relative!)
cp'=hck'cctt
cctt''
EnergyE=hω
Momentumcp=hck
Mc2
ωm=49ω1
76543210-1-2-3-4-4-6m
36
251694
(a) Einstein-Planck Dispersion
(b) DeBroglie-Bohr Dispersion
E'=hω'E2 - c2p2 =(Mc2)2
photon: M=0E = c p
E = p2/2M
E = B m2
tachyon:
Bohr - Schrodinger Dispersion
Einstein - Planck Dispersion
ω BS (k) = k 2
2M gives: v phase=
ω BS
k= k
2M
and: vgroup=dω BS
dk= kM
29Tuesday, April 3, 2012
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=dkdω
kω
Group velocity u=Vgroup is a differential quantity unaffected by origin shift.
But, Phase velocity =Vphase is greatly reduced by deleting Mc2 from E=ω.
It slows from Vphase=c2/u to a sedate sub-luminal speed of Vgroup/2.
Bohr-Schrodinger (BS) approximation throws out Mc2
E =Mc2
1− u2 / c2= Mc2 coshρ = Mc2 1+ sinh2 ρ = Mc2( )2 + cp( )2
E = Mc2( )2 + cp( )2⎡
⎣⎢
⎤
⎦⎥1/2
≈ Mc2 + 12M
p2 BS−approx⎯ →⎯⎯⎯⎯12M
p2
The BS claim: may shift energy origin (E=Mc2, cp=0) to (E=0, cp=0). (Frequency is relative!)
cp'=hck'cctt
cctt''
EnergyE=hω
Momentumcp=hck
Mc2
ωm=49ω1
76543210-1-2-3-4-4-6m
36
251694
(a) Einstein-Planck Dispersion
(b) DeBroglie-Bohr Dispersion
E'=hω'E2 - c2p2 =(Mc2)2
photon: M=0E = c p
E = p2/2M
E = B m2
tachyon:
Bohr - Schrodinger Dispersion
Einstein - Planck Dispersion
ω BS (k) = k 2
2M gives: Vphase=
ω BS
k= k
2M
and: Vgroup=dω BS
dk= kM
30Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
31Tuesday, April 3, 2012
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(a) Hamiltonian
Momentum p
PP′
P′′
-L-L′-L′′
L(q,q)
Velocity u=q
QQ′Q′′
-H
-H′
-H′′
HH′
H′′LL′L′′
H
H′
H′′
slope:
slope:
∂H∂p
= q= u
∂L∂q
= p
(b) LagrangianH(q,p)
radius = Mc2
O
O
Light cone u=1=chas infiniteH
and zero L
Fig. 5.1. Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian
Deriving relativistic quantum Lagrangian-Hamiltonian relations
dΦ = kdx − ω dt= −µ dτ = -(Mc2/) dτ. dτ = dt √(1-u2/c2)=dt sech ρ
Differential action: is Planck scale times differential phase: dS = dΦ
For constant u the Lagrangian is: L =−µτ = -Mc2√(1-u2/c2)= -Mc2sech ρ = -Mc2
Start with phase and set k=0 to get product of proper frequency and proper timeΦ µ = Mc2/ τ
dS = Ldt = p·dx − H·dt = k·dx − ω ·dt = dΦ
cosσ
32Tuesday, April 3, 2012
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(a) Hamiltonian
Momentum p
PP′
P′′
-L-L′-L′′
L(q,q)
Velocity u=q
QQ′Q′′
-H
-H′
-H′′
HH′
H′′LL′L′′
H
H′
H′′
slope:
slope:
∂H∂p
= q= u
∂L∂q
= p
(b) LagrangianH(q,p)
radius = Mc2
O
O
Light cone u=1=chas infiniteH
and zero L
Fig. 5.1. Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian
Deriving relativistic quantum Lagrangian-Hamiltonian relations
dΦ = kdx − ω dt= −µ dτ = -(Mc2/) dτ. dτ = dt √(1-u2/c2)=dt sech ρ
Differential action: is Planck scale times differential phase: dS = dΦ
For constant u the Lagrangian is: L =−µτ = -Mc2√(1-u2/c2)= -Mc2sech ρ = -Mc2
Start with phase and set k=0 to get product of proper frequency and proper timeΦ µ = Mc2/ τ
cosσ
dS = Ldt = p·dx − H·dt = k·dx − ω ·dt = dΦ
33Tuesday, April 3, 2012
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(a) Hamiltonian
Momentum p
PP′
P′′
-L-L′-L′′
L(q,q)
Velocity u=q
QQ′Q′′
-H
-H′
-H′′
HH′
H′′LL′L′′
H
H′
H′′
slope:
slope:
∂H∂p
= q= u
∂L∂q
= p
(b) LagrangianH(q,p)
radius = Mc2
O
O
Light cone u=1=chas infiniteH
and zero L
Fig. 5.1. Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian
Deriving relativistic quantum Lagrangian-Hamiltonian relations
dΦ = kdx − ω dt= −µ dτ = -(Mc2/) dτ. dτ = dt √(1-u2/c2)=dt sech ρ
Differential action: is Planck scale times differential phase: dS = dΦ
For constant u the Lagrangian is: L =−µτ = -Mc2√(1-u2/c2)= -Mc2sech ρ = -Mc2
Start with phase and set k=0 to get product of proper frequency and proper timeΦ µ = Mc2/ τ
cosσ
dS = Ldt = p·dx − H·dt = k·dx − ω ·dt = dΦ
34Tuesday, April 3, 2012
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(a) Hamiltonian
Momentum p
PP′
P′′
-L-L′-L′′
L(q,q)
Velocity u=q
QQ′Q′′
-H
-H′
-H′′
HH′
H′′LL′L′′
H
H′
H′′
slope:
slope:
∂H∂p
= q= u
∂L∂q
= p
(b) LagrangianH(q,p)
radius = Mc2
O
O
Light cone u=1=chas infiniteH
and zero L
Fig. 5.1. Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian
Deriving relativistic quantum Lagrangian-Hamiltonian relations
dΦ = kdx − ω dt= −µ dτ = -(Mc2/) dτ. dτ = dt √(1-u2/c2)=dt sech ρ
Differential action: is Planck scale times differential phase: dS = dΦ
For constant u the Lagrangian is: L =−µτ = -Mc2√(1-u2/c2)= -Mc2sech ρ = -Mc2
Start with phase and set k=0 to get product of proper frequency and proper timeΦ µ = Mc2/ τ
dS = Ldt = p·dx − H·dt = k·dx − ω ·dt = dΦ
cosσ
L= p· x − H = p·u − H...with Poincare invariant:
35Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
36Tuesday, April 3, 2012
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Fig. 5.2 “True” paths carry extreme phase and fastest clocks. Light-cone has only stopped clocks.
Fig. 5.3 Quantum waves interfere constructively on “True” path but mostly cancel elsewhere.
Feynman’s flying clock and phase minimization
37Tuesday, April 3, 2012
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Group vs. phase velocity and tangent contacts Reviewing “Sin-Tan Rosetta” geometry
How optical CW group and phase properties give relativistic mechanics Three kinds of mass (Einstein rest mass, Galilean momentum mass, Newtonian inertial mass)
What’s the matter with light? Bohr-Schrodinger (BS) approximation throws out Mc2
Deriving relativistic quantum Lagrangian-Hamiltonian relationsFeynman’s flying clock and phase minimization
Geometry of relativistic mechanics
Relativistic Classical and Quantum Mechanics
38Tuesday, April 3, 2012
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ρ
ρHHaammiillttoonniiaann ==HH ==BB ccoosshh ρ
--LLaaggrraannggiiaann
--LL ==BB sseecchh ρRReesstt
EEnneerrggyy
BB ==MMcc22
HH
BB
--LL
VVeelloocciittyy aabbeerrrraattiioonn
aannggllee φ
33−
433221100--11
4
--22
BB ccoosshh ρ
BB ee--ρcp
H=E
BB
BB ee++ρ
BB sseecchh ρ
22
11
BBlluuee sshhiiffttRReedd sshhiifftt
(a) Geometry of relativistic transformationand wave based mechanics
(b) Tangent geometry (u/c=3/5)
(c) Basic construction given u/c=45/53
u/c =3/5u/c =1
cp =3/4
H =5/4
-L =4/5
cp =45/28
H =53/28
-L=28/45
e-ρ=1/2e-ρ=2/7
(d) u/c=3/5
11
1 1
bb--cciirrccllee
g-circle
p-circle
Fig. 5.5 Relativistic wave mechanics geometry. (a) Overview.
(b-d) Details of contacting tangents.
39Tuesday, April 3, 2012