18
R. N. Mohapatra
defined in Newtonian mechanics as
p = m v, c is the speed of light which
enters many equations in physics and begins to play a major role. Max Planck
is believed to have said “The velocity of light is to the Theory of Relativity as the
elementary quantum of action is to the Quantum Theory: it is its absolute core.”.
1
For a particle at rest, Einstein’s formula for energy says that there is an
equivalence between mass and energy given by the formula E = mc
2 , (which
nowadays appears on t-shirts). It is at the core of many applications of nuclear
energy. In nuclear reactors, when one nucleus transmutes to one nucleus in
beta decay or to two nuclei, the difference in their mass gets transformed into
energy and is available as nuclear energy, all due to the E = mc
2 relation of
Einstein.
The difference between the two formulas for energy above, the one for slow
moving particles (non-relativistic) and the other one for fast moving particles
(relativistic), may not appear so fundamental at first sight, but there is a hidden
aspect of this which had far reaching implications. The point is that in the
first non-relativistic formula the energy is always positive as we commonly
understand—however, in the second formula, which involves energy square,
the energy E can be positive or negative. This is because of the peculiar
mathematical property of a square root of a positive number, which can be
either positive or negative, i.e. the square of both +2 or −2 is +4.
What is the meaning of the negative value of energy? When we run, we
have energy and it is always a positive number. No moving object has negative
energy. But it appears that in the theory of relativity, there are states with
negative energy! How to make sense of the negative energy? Should we just
discard the negative value as being unphysical? That is certainly one possibility,
but that would be rather arbitrary since it comes out of a perfectly sensible
equation in relativity. It would appear that we have no choice but to accept
these states as real states in nature.
The true significance of energy being negative was first realized by British
physicist Paul Andre Maurice Dirac in 1928. Dirac was a deep thinking, quiet
man. This author has been in lunch gatherings at Stonybrook where Dirac
was a visitor in the early 1970s. At the end of the lunch, the person who
talked the least was Dirac. In 1928, Dirac was trying to construct a theory
of the electron within the relativistic framework (for fast moving electrons)
generalizing the work of Erwin Schroedinger. Schroedinger wrote down the
equation, describing quantum mechanical motion of slower moving particles.
1 Incidentally elementary quantum of action (which is defined as energy times time) is denoted by the
symbol h and is the smallest quantum of action of an atomic system. This is known as Planck’s constant.
All actions are integral multiples of this fundamental quantum.
R. N. Mohapatra
defined in Newtonian mechanics as
p = m v, c is the speed of light which
enters many equations in physics and begins to play a major role. Max Planck
is believed to have said “The velocity of light is to the Theory of Relativity as the
elementary quantum of action is to the Quantum Theory: it is its absolute core.”.
1
For a particle at rest, Einstein’s formula for energy says that there is an
equivalence between mass and energy given by the formula E = mc
2 , (which
nowadays appears on t-shirts). It is at the core of many applications of nuclear
energy. In nuclear reactors, when one nucleus transmutes to one nucleus in
beta decay or to two nuclei, the difference in their mass gets transformed into
energy and is available as nuclear energy, all due to the E = mc
2 relation of
Einstein.
The difference between the two formulas for energy above, the one for slow
moving particles (non-relativistic) and the other one for fast moving particles
(relativistic), may not appear so fundamental at first sight, but there is a hidden
aspect of this which had far reaching implications. The point is that in the
first non-relativistic formula the energy is always positive as we commonly
understand—however, in the second formula, which involves energy square,
the energy E can be positive or negative. This is because of the peculiar
mathematical property of a square root of a positive number, which can be
either positive or negative, i.e. the square of both +2 or −2 is +4.
What is the meaning of the negative value of energy? When we run, we
have energy and it is always a positive number. No moving object has negative
energy. But it appears that in the theory of relativity, there are states with
negative energy! How to make sense of the negative energy? Should we just
discard the negative value as being unphysical? That is certainly one possibility,
but that would be rather arbitrary since it comes out of a perfectly sensible
equation in relativity. It would appear that we have no choice but to accept
these states as real states in nature.
The true significance of energy being negative was first realized by British
physicist Paul Andre Maurice Dirac in 1928. Dirac was a deep thinking, quiet
man. This author has been in lunch gatherings at Stonybrook where Dirac
was a visitor in the early 1970s. At the end of the lunch, the person who
talked the least was Dirac. In 1928, Dirac was trying to construct a theory
of the electron within the relativistic framework (for fast moving electrons)
generalizing the work of Erwin Schroedinger. Schroedinger wrote down the
equation, describing quantum mechanical motion of slower moving particles.
1 Incidentally elementary quantum of action (which is defined as energy times time) is denoted by the
symbol h and is the smallest quantum of action of an atomic system. This is known as Planck’s constant.
All actions are integral multiples of this fundamental quantum.