Lorentz Group derivable from Heisenberg's Uncertainty CommutatorsWe all know how quantum mechanics was developed, and how Einstein created his special theory of relativity. Heisenberg wrote down one set of commutation relations for three space and three momentum variables.
Einstein's special relativity is based on the mathematics of the Lorentz group developed earlier by Lorentz and Poincaré.
If I tell you I can derive the Lorentz group from Heisenberg's commutation relations, you will tell me to go to heaven. If I tell you Paul A. M. Dirac did so in 1963, you might like to hear about it. Let me tell you the story.
- Heisenberg was interested in the Poisson brackets in classical
mechanics. We all know what he did with them. So was
Paul A. M. Dirac. He was interested in extending the Poisson brackets
to Einstein's Lorentz-covariant world. See Dirac's in 1949 paper
It is interesting to note that Dirac kept using the word "Poisson brackets" for the uncertainty relations.
Yes. Dirac was able to construct the Lorentz group starting from the Poisson brackets. The question is when and how?
Dirac met Heisenberg frequently in Europe before 1939. Did
he figure out this while he was talking with Heisenberg? The
answer is No.
It was not until 1963 that Dirac published his paper on the derivation of the O(3,2) symmetry from two harmonic oscillators. Click here for the paper.
What happened between 1927 and 1963? The role of Heisenberg's commutation relations for quantum mechanics is well known. In addition, the Heisenberg relations lead to the following algebras.
- The commutation relations for the generators of the
rotation group are derivable from the Heisenberg relations
It is possible to construct the Lorentz group by adding
the time coordinate to this set of commutation relations.
See how this happens.
- It is possible to convert the Heisenberg relations to the algebra of harmonic oscillators, consisting of step-up and step-down operators. These up and down operators can be used for the particle creation and annihilation operators the Fock space. This aspect is also well known. Click here for a photo of Vladimir Fock talking to Dirac in 1962.
- The commutation relations for the generators of the rotation group are derivable from the Heisenberg relations It is possible to construct the Lorentz group by adding the time coordinate to this set of commutation relations. See how this happens.
- One oscillator algebra consists of two operators, namely step-up and
step-down operators. For two oscillators, there are
two step-up and two step-down operators. There are thus four operators.
There are thus sixteen bilinear operators. Click here
for detailed calculations.
Using ten of these combinations, Paul A, M. Dirac composed the following mathematical poem.
There are ten operators, which form a closed set of commutation relations. Dirac in 1963 observed that this set is the same as the Lie algebra for the group of Lorentz transformations applicable to three space variables and two time variables. Click here for his 1963 paper.
My photo with Roy Galuber (2001),
with Horace Yuen (1997).
Dirac with his wife (1963). Did you know Mrs. Dirac was Wigner's younger sister?
Click here for a story.
- In the above table, three are L operators. They serve as
the generators of rotations in the 3-dimensional space. Three K
operators are for the Lorentz boost with respect to one of the time
variables. Three Q are for the second time variable. The
S operator couples these two sets of the four-dimensional
This group applicable to the five dimensional space is called the 3+2 deSitter group or O(3,2). This group plays a role in Einstein's general relativity.
This deSitter group is also useful in down-to-earth physics. It can serve as two coupled Lorentz groups with with variable masses. This group is also applicable to the Stokes parameters and Poincaré sphere in polarization optics.
Since we need one Lorentz group in the real world, the other Lorentz group can serve as Feynman's rest of the universe.
- While there are 16 possible bilinear combinations, Dirac used only ten of
them. What is the role of the remaining six operators? This is an interesting
question. Click here for a published paper.
Physics faculty photo of 1963 at the University of Maryland. I am the youngest person in this photo, a perfect candidate for
Dirac's servant in 1962.
I became an assistant professor at the University of Maryland after receiving my degree in 1961. In the fall of 1962, Dirac spent about ten days at UMD. He came at the invitation of John S. Toll who was the chairman of the department at that time. Toll assigned me as Dirac's servant, and I was able to talk to him.
Michelangelo's portrait of
Nicodemus or himself.
Toll and Wheeler at UMD(2000).
Toll, Mrs. Toll, Wigner, and Kim
at UMD (1986).
- In 1962, I was not happy with what was going on in the physics world.
Click here to see how unhappy I was. Indeed,
my contact with Dirac was like Nicodemus asking Jesus what to do.
The story of Nicodemus is in the Gospel of John in the New Testament. Click here for the Wikipedia story. Nicodemus was a member of the Pharisees (upper-class Jews), but he did not like the society in which he belonged. He made a secret visit to Jesus to ask for wisdom. Jesus told him to be BORN AGAIN.
I am not the first one to identify himself with Nicodemus. Michelangelo thought he was a new Nicodemus. He did not like the society dictated by the church. His role in the Renaissance is well known. In his Florentine Pieta, Michelangello constructed a statue of Nicodemus, but it is agreed that it was his own statue.
- Finally, I would like to use this space to express my gratitude to
John S. Toll. Like most of you, my life as a physicist was not always
easy, mostly due to my colleagues who did not like what I was doing
in physics. For instance, a number of distinguished persons told
Toll to reduce my position when I was organizing a conference for
Eugene Wigner in 1988. Yet, he firmly supported me.
John Toll came from a distinguished family. His father served as one of the lawyers at the Nuremberg Trials. He came to the University of Maryland as the chairman of the Department in 1953 after receiving hid degree at Princeton. His advisor was John A. Wheeler.
Toll spent one summer in Paris while he was a graduate student. There, he used to go out with a girl named Jacqueline Bouvier, who became Jacqueline Kennedy in 1953. He used to become very happy when Wigner came to Maryland at my invitation during the period 1984-88. Click here for a photo.
Needless to say, the best thing he did for me was to assign me as a servant to Dirac in 1962.
Click here for more photos.
- The University of Maryland is about 16 km north of the White House. For
this reason, I often update my Washington page. This city has many statues
of those who made contributions to the United States, including Washington,
Lincoln, Jefferson, Franklin Roosevelt, Martin Luther King, and many others.
Among the statues of scientists are those of Albert Einstein and Guglielmo Marconi. Who was Marconi?
Click here for my Washington page. The city becomes exceptionally bright during the cherry blossom season
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