Page 1 of Finding Particles Page 2

You can imagine these vectors like *arrows*,
pointing in a certain direction.
Since our world has three dimensions
(left-right, forward-backward, up-down)
a vector has *three vector components*,
describing the speed in each
of the three fundamental directions.

In physics you normally denote them with
**x, y, and z**, and the overall vector
with a small arrow over its symbol:

**Speed, momentum and kinetic energy**

In physics all objects have a really
*serious speed limit*, the speed of light!
Nothing can move faster than light (in vacuum),
called ``c''.
That's a mind-boggling large speed of
ca. 300,000 km per second,
but still it's the absolut limit.
Therefore all moving particles with high speed,
like particles in HEP,
have their velocity measured as ratio over
the speed of light.
For this *vector* we use
the greek symbol ``beta'':

It is a well-known fact in physics that speed means
the object has energy.
That is the reason,
why an abrupt stop of a moving car with a tree
causes damage to both.
The amount of ``movement energy'',
called *kinetic energy*,
is a very important number.
In non-relativistic mechanics
they are connected by:

Since in many physical processes, the so called ``momentum'' (product of mass and velocity) is conserved, it is normally written as:

The mathematical operation of muliplying two vectors, resulting in a single number (also called ``scalar'') is called ``scalar product''. For our vectors it is defined as:

As you can see, multiplying a vector with itself results in the ``pythagoras equation'', giving us theIn our little example of driving in a car on the freeway, this transformation connects the vector of your car seen by observer on a bridge

Both obervers see the same **length of the vector**
although the components of v **are different**.
For the observer on the bridge you are moving *away*,
but for the police man you are coming *closer*.

This means,
all physical processes calculated in each reference frame,
will give the same results,
if they use **vector lengths** in their mathematical formulation,
like the kinetic energy does.

**Vectors in special relativity**

This fundamental approach of physics

- Find all
*valid transformation*between reference frames, and - formulate all laws with
*invariant values*under these transformations including vector lengths.

Physicist distinguish those vectors from normal vectors by using a

The

*Note the - sign* in the last equation.
This product value stays constant
under all legal transformation (the ``Lorentz transformations'').

**Combining momentum and energy**

Besides all those widely known consequences of this theory,
like slower ticking clocks in moving rockets
and shortened lengths of speeding cars,
the *momentum vector* also has to become a vector
with four components.
Einstein discovered that this ``zero-th'' component
has to be the total energy of the moving particle,
and the *length of the vector* equals
the **mass** of the object.
This results in the most famous equation of this theory:

The first equation is obtained by using the ``new'' scalar product
for 4-vectors.

The second equation is an approximation for very slow particles
and shows that the normal equation for the kinematic energy is
recovered, *but* there is an additional term
that connects *mass* with *energy*.

Since the length of the 4-vector is again constant under all
allowed transformation, this mass is called the ``invariant mass''.

We in the HEP field use mass units that show
this equality of mass and energy.
We also ``redefine'' the world by using ``natural units'',
in which ``c'' (the speed of light) *equals 1*.
The mass of an object is measured in **eV**.
1 eV is the energy an *object with the charge of the electron*
receives or looses as kinetic energy,
when it passes the *voltage difference of 1 V*.
This unit was chosen, since all particle accelerators work with
high voltage differences to accelerate charged particles.

1 eV is equal to 1.8 10**-36 kg...

Page 1 of Finding Particles Page 2

Updated: 28. November 1995