feynman and dirac bonded by squeezed states
TRANSCRIPT
![Page 1: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/1.jpg)
Feynman and Dirac
bonded by
Squeezed States
Y. S. Kim
University of Maryland
U.S.A.
![Page 2: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/2.jpg)
Dirac and Feynman in Poland (1962)
• Dirac: construct a
beautiful mathematics.
• Feynman: mathematical
instrument that will
produce numbers which
can be compared with
what we observe in the
real world.
![Page 3: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/3.jpg)
How?
• 2-by-2 matrices.
rotations and
squeeze
• 2-d Gaussian
distribution
![Page 4: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/4.jpg)
Audience
with Dirac
in 1962
Like
Nicodemus
asking
Jesus
![Page 5: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/5.jpg)
I asked Dirac
what I should
do in Physics.
Study Lorentz
covariance.
Americans
should study
more about
this subject.
![Page 6: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/6.jpg)
Five Books of Dirac
• First Book. 1927 paper on time-energy
uncertainty – c-number
• Second Book. 1945 paper on harmonic
oscillators. Gaussian time distribution.
• Third Book. 1949 paper on light-cone
coordinate system
• Fourth Book: 1963 paper on two-oscill.
system-– two mode squeezed states.
• Fifth Book. To be written.
![Page 7: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/7.jpg)
Why do we need the Fifth Book?
• Dirac’s papers are like poems, but contain
no figures. Translate those poems into
cartoons.
• Those papers lack continuity. Dirac does
not make reference to his earlier papers.
• Dirac does not make difference
between time and time interval.
![Page 8: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/8.jpg)
Feynman‟s Interpretation of
Physics
The adventure of our
science of physics is a
perpetual attempt to
recognize that the different
aspects of nature are really
different aspects of
the same thing.
![Page 9: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/9.jpg)
• Feynman wrote 150 papers.
• He wanted to pack them into one paper.
• IWe can combine three of them into one
paper..
1969 paper on partons.
1971 paper on covariant oscillators
1972 book. Rest of the universe.
Concept of entanglement.
It then becomes the Fifth Book of
Paul A, M Dirac
![Page 10: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/10.jpg)
1970 Feynman’s Crazy Talk
![Page 11: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/11.jpg)
Feynman said!
• For Scattering problems, use Feynman
diagrams.
• For bound-state problems, use harmonic
oscillators.
• However, there were no well-known Lorentz-
covariant oscillators in 1970.
• People said Feynman was absolutely crazy.
• But, I became excited about what he said,
and studied his physics systematically since
then. I also became crazy.
![Page 12: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/12.jpg)
1965. Feynman diagrams do
not guarantee localization.
![Page 13: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/13.jpg)
Based on Feynman’s 1970 Talk
![Page 14: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/14.jpg)
Feynman et al. acknowledge the
existence of time separation between
quarks inside the hadron
They choose not to do anything about it,
because they do not know what to do.
They fail to mention earlier works on this
subject.
![Page 15: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/15.jpg)
Main Issue
Time
Separation
Entanglement
Hidden
Variable
![Page 16: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/16.jpg)
Yukawa (1953), Markov (1953),
Man’ko (1965), Fujimura (1970)
![Page 17: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/17.jpg)
I am known as a person who
maintains webpages
http://ysfine.com
for conference catalog
http://ysfine.com/princeton
for beautiful campus
http://ysfine.com/einstein
for Einstein
![Page 18: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/18.jpg)
I look very old these days.
• Did I meet Einstein?
• No. I was too young
• I met some people who met
him.
![Page 19: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/19.jpg)
Orren Jack Turner
1947 1961
![Page 20: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/20.jpg)
Hidden behind Einstein and Bohr
![Page 21: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/21.jpg)
Niels Bohr
• Bohr had a great respect for Einstein.
Whenever he mentions “space,” he
adds “time „ in his philosophical
writings.
• However, the proton in the hydrogen
atom is sitting at the center of the
absolute frame.
![Page 22: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/22.jpg)
Einstein perhaps had a picture
of Lorentz ontraction, but never
mentioed
![Page 23: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/23.jpg)
Bohr radius: space separation
• Time
separation
was a
hidden
variable to
them.
![Page 24: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/24.jpg)
Evolution of Environment
• All men are
created equal.
• How about all
women?
• How about his
servants?
![Page 25: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/25.jpg)
Jefferson’s Monticello
![Page 26: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/26.jpg)
Evolution of hydrogen atom
• Hydrogen
becomes a
hadron, a
bound state of
quarks.
• Speed of
proton close to
that of light.
![Page 27: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/27.jpg)
Hydrogen Atom becomes the
Hadron (bound state of two quarks)
in the Lorentz-covariant world.
![Page 28: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/28.jpg)
Time
separation
well defined.
Dirac 1927
and 1945
![Page 29: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/29.jpg)
Space-time asymmetry? Dirac
should have talked to Wigner
![Page 30: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/30.jpg)
Dirac
1945
![Page 31: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/31.jpg)
Physics
or
Science
Fiction
![Page 32: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/32.jpg)
Let us invite Feynman
![Page 33: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/33.jpg)
Covariant Bound States
(Standing Waves)
• Bound States: Hydrogen Atom or
Harmonic Oscillators.
• Feynman chooses osc. wave functions to understand the covariant world.
• Hadron consisting of two quarks. Overall coordinate and space-time separation.
![Page 34: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/34.jpg)
Feynman said
![Page 35: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/35.jpg)
Feynman’s One Equation
In his 1971 paper,
Feynman wrote down
one partial differential
equation to cover
both comets and
planets.
Where is the
Squeezed State?
![Page 36: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/36.jpg)
Feynman’s One Equation
![Page 37: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/37.jpg)
![Page 38: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/38.jpg)
Hadrons – Running waves
Quarks inside – Harmonic Oscillator
![Page 39: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/39.jpg)
![Page 40: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/40.jpg)
Feynman was successful in
solving the mysteries of Regge
poles, but totally failed in
providing Lorentz-covariant
solutions of his own equation for
harmonic oscillators.
From the Lorentzian point of
view, his paper is total mess.
![Page 41: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/41.jpg)
Dirac and Feynman in Poland (1962)
When Dirac told me
American physicists
should spend more
time on Lorentz
covariance, he was
talking about
Feynman.
![Page 42: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/42.jpg)
When Dirac was talking to me
about American physicists in
1962, he was talking about
Feynman.
Thus, one way to fix Feynman’s
problem was to import Dirac’s
work.
![Page 43: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/43.jpg)
![Page 44: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/44.jpg)
Yukawa (1953), Markov (1953),
Man’ko (1965), Fujimura (1970)
![Page 45: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/45.jpg)
![Page 46: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/46.jpg)
Feynman’s One Equation
In order to handle
the differential
equation
Feynman
wrote down, we
have to import
things from both
Wigner and Dirac.
![Page 47: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/47.jpg)
Two famous brothers in law
![Page 48: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/48.jpg)
Running waves and Standing Waves
![Page 49: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/49.jpg)
Feynman said
![Page 50: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/50.jpg)
Time
separation
![Page 51: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/51.jpg)
Wigner’s Little Group
• The space-time symmetry inside the hadron is
smaller than the Lorentz group. It is isomorphic
to O(3), three-dimensional rotation group.
![Page 52: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/52.jpg)
After fixing up Feynman’s
oscillators
We can assert that Dirac‟s Five Books
are for the real world.
![Page 53: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/53.jpg)
![Page 54: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/54.jpg)
Parton Picture
Coherence
to
Incoherence
![Page 55: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/55.jpg)
Lorentz-covariant
Picture of Feynman’s
Parton Model
![Page 56: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/56.jpg)
Two different aspects
(quarks and partons) of nature
coming from one thing!
![Page 57: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/57.jpg)
Register
ownership
with
authorities
Phys. Rev.
Letters
![Page 58: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/58.jpg)
Still uncomfortable with
the time separation?
![Page 59: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/59.jpg)
Same math
Physics can
be
transferred
![Page 60: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/60.jpg)
Theory of
Everything
in Physics
![Page 61: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/61.jpg)
![Page 62: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/62.jpg)
Dirac
1962
Two-
oscillator
system
Two-mode
squeezed
states
![Page 63: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/63.jpg)
![Page 64: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/64.jpg)
![Page 65: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/65.jpg)
The same formula leads to
Top: Squeezed State
Bottom: Entangled State
![Page 66: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/66.jpg)
Feynman was a Kantianist,
so was Einstein.
The adventure of our
science of physics is a
perpetual attempt to
recognize that the
different aspects of
nature are really different
aspects of the same thing.
![Page 67: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/67.jpg)
Entropy Problem
• First, let us look at what he says about the density
matrix. In his 1973 book, Feynman says
![Page 68: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/68.jpg)
Here again, Feynman talks like
a philosopher
• Here again, we can use harmonic
oscillators to see what he really had in
mind.
• In his original paper he uses a single
oscillator.
• We should use coupled oscillators to
decode what he was saying.
![Page 69: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/69.jpg)
Rest of the universe --
Entanglement
• X variable: we do
physics with.
• Y variable: we do not
observe. – rest of the
universe.
Two worlds are entangled.
![Page 70: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/70.jpg)
![Page 71: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/71.jpg)
![Page 72: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/72.jpg)
![Page 73: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/73.jpg)
![Page 74: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/74.jpg)
![Page 75: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/75.jpg)
![Page 76: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/76.jpg)
![Page 77: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/77.jpg)
![Page 78: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/78.jpg)
![Page 79: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/79.jpg)
![Page 80: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/80.jpg)
Back to Space-time physics
• Time separation variable between the
quarks exists, but they are not
observed.
• It indeed belongs to Feynman‟s rest of
the universe.
• First photon becomes z
• Second photon becomes t.
![Page 81: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/81.jpg)
Register
ownership
with
authorities
Phys. Rev.
Letters
![Page 82: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/82.jpg)
![Page 83: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/83.jpg)
![Page 84: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/84.jpg)
![Page 85: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/85.jpg)
Effects of hidden variables
• Time separation
as a hidden
variable leads to
an increase in
uncertainty.
• The real issue:
what is the
variable hidden
in the small
circle.?
![Page 86: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/86.jpg)
We started writing papers on
Feynmanism from 1973
![Page 87: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/87.jpg)
2003
![Page 88: Feynman and Dirac bonded by Squeezed States](https://reader031.vdocuments.us/reader031/viewer/2022012011/613d7167736caf36b75d63b9/html5/thumbnails/88.jpg)
Thank you for listening!
Visit
http://ysfine.com
and click on
Show and Tell
For interesting stories