# Further Contents of *E = mc*^{2}

Albert Einstein was born in Ulm (Germany)
on March 14, 1879. However, his family moved to Munch when he was
an infant. There, his family set up a profitable business of
manufacturing electric appliances. However, due to the AC revolution,
their business became bankrupt.
^{2}

- Einstein's family had to move to a small town near Milan (Italy).
In 1895, at the age of 16, Albert went to Switzerland. There, he finished
his high-school education. Two year later in 1897, Einstein entered Zürich
Federal Polytechnic for his 4-year college program. This school is known
today as ETH
(Eidgenössische Technische Hochschule).
*Einstein in 1904 or 1905*

Birth Place of *E = mc*^{2}In this house, during the years

1903-1905, Albert Einstein created his fundamental treatise on relativity theory. - While working for the patent office, Einstein pursued his physics research
and completed his special theory of relativity. He did not pull out
this revolutionary theory from the blue sky.
- During his high-school years, he was immensely interested in the
philosophy of Kant who said things could look differently to different
observers depending on where they are or how they think.
Click here for his philosophical
background.
- He was quite interested in the mathematics of Lorentz transformations
developed earlier by Lorentz and Poincaré. He also learned from
Hermann Minkowski that Maxwell's equations are covariant under Lorentz
transformations.
- He noticed the velocity addition rule is cumbersome in the Lorentzian
regime. He thus tried transformation laws for the momentum, and
came to the conclusion that the momentum and energy form a four-vector,
just like the space-time four-vector.
While

*t*^{2}- x^{2}- y^{2}- z^{2}is an invariant quantity in the Lorentzian world, Einstein came to the conclusion that

*E*^{2}- p_{x}^{2}- p_{x}^{2}- p_{x}^{2}= mass^{2}.This is what Einstein's

*E = mc*is all about.^{2}

This completes the energy-momentum relation for particles in the Lorentz-covariant world.

- During his high-school years, he was immensely interested in the
philosophy of Kant who said things could look differently to different
observers depending on where they are or how they think.
Click here for his philosophical
background.

## What happens when the particle has internal space-time structures ?

- Bohr and Einstein met occasionally to discuss physics. Einstein was interested
in how things appear to moving observers. Bohr was interested in the electron
orbit of the hydrogen atom. Did they ever talk about how the electron orbit
looks to moving observers? This is still the most outstanding problem in
physics.
Isaac Newton had a similar problem when he formulated his law of gravity. He wrote down the formula for the inverse square law between two point particles. It then took him 20 years to complete the law for extended objects. He had to invent a new mathematics known today as "Integral Calculus," as shown in the following figure.

The issue is then whether we have to invent a new mathematics or we can find suitable mathematical devices from the existing literature in mathematics. Indeed, in 1939, Eugene Wigner formulated the concept of the little groups which dictate the internal space-time symmetries of the particles in the Lorentz covariant world, as illustrated in this figure.

This aspect of Wigner's work is yet to be recognized. - In 1939, Wigner constructed subgroups of the Lorentz group whose
transformations leave the momentum of a given particle invariant.
- If the particle is massive, there is a Lorentz frame in which
the particle is at rest. In this frame, the momentum is invariant
under three-dimensional rotations. Thus, the little group is O(3) or
the three-dimensional rotation group. This indeed tells the spin
degree of freedom.
- In 1939, Wigner observed that the little group for massless
particles is isomorphic to E(2) or two-dimensional Euclidean group
with one rotational and two translational degrees of freedom. It
is easy to associate the rotational degree with the helicity of
the massless particle. What physics is associated with the
translations. This question was not completely settled until
1987. These two translational degrees of freedom collapse into
one gauge degree of freedom.
Click here for details.
Einstein's the energy-momentum relation for the massive particle becomes the same as that for massless particle when its momentum is much larger than the mass. Likewise the O(3)-like little group for massive particle becomes the E(2) like little group when the momentum of the particle is much larger than the particle mass. Click here for a published article.

*Einstein's Genealogy*Massive/Slow between Massless/Fast Energy

MomentumE=p ^{2}/2mEinstein's

E=(m^{2}+ p^{2})^{1/2}E=p Helicity

Spin, GaugeS _{3}

S_{1}S_{2}Winner's

Little GroupHelicity

Gauge Trans.Click here for a photo. Build

YOUR OWN HOUSE!You are then invited to build your own house. How are you going to build your own house? With what? In order to build a house consistent with it neighborhood is to find a Lorentz-covariant entity which takes different forms for slow and fast particles.

## See next page.

portrait by Bulent Atalay (1978)

- In 1939, Wigner constructed subgroups of the Lorentz group whose
transformations leave the momentum of a given particle invariant.