# Dirac and Squeezed States

Paul A. M. developed the concept of squeezed states in two different ways.
1. Space-time Squeeze. By introducing his light cone coordinate system he observed that the Lorentz boost is a squeeze transformation.

2. Two-photon States. By constructing an algebra for two coupled harmonic oscillators, he worked out the symmetry of two-photon states.

Let us discuss these issues in detail.

## 1. Space-time Squeeze

• According to Einstein's special relativity, Lorentz boosts leave

z2 - t2

invariant. In 1949 [ Rev. Mod. Phys. 21 , 392 (1949)], Dirac noted that this quantity can be written as

(z + t)(z - t) = constant.

If (z + t) becomes large, (z - t) should become small. This is what the space-time squeeze is all about.

• Earlier in 1927 [Proc. Roy. Soc. (London) A114, 243], Dirac pointed out there is an uncertainty relation between time and energy variables. Unlike Heisenberg's uncertainty, there are no excitations along the time axis. Dirac called it the c-number time-energy uncertainty relation.

• In 1945 [Proc. Roy. Soc. (London) A183 , 284], Dirac attempted to use harmonic oscillators to construct representations of the Lorentz group. However, he did not give a physical interpretation to the time variable appearing his Gaussian Gaussian form.

• In 1971, Feynman and his students observed that that observed hadronic mass spectra can be interpreted in terms of three-dimensional harmonic oscillators. However, they fail to construct Lorentz-covariant wave functions. For instance, they choose to ignore the time variable in the wave function simply because they do not know how to handle it. This is not the right way to do physics.

• In 1973, Kim and Noz noted that Yukawa in 1953 had constructed the normalizable oscillator wave functions, and discuss the calculation of the of the proton form factor by Fujimura, Kobayashi, and Namiki who obtained the dipole cutoff of the form factor.

• Indeed, it is possible to combine Dirac's idea and Feynman's idea to solve some of the outstanding problems in high-energy physics. Click here for a story.

## 2. Dirac's Two-photon States

• In 1963 [ J. Math. Phys. 4 , 901 (1963) ], Dirac published a paper containing ten bilinear combinations of the step-up and step-down operators. Click here for their expressions.

 Formulas from Dirac's 1963 paper
1. Dirac noted that these operators satisfy the commutations relations for the generators of the O(3,2) de Sitter group, which is the Lorentz group applicable to the three space dimensions and two time dimensions.

2. Dirac acknowledged in his paper that he was told by a younger physicist that those ten bilinear forms can also serve as the generators of the four-dimension symplectic group applicable to the phase space consisting of two coordinate and two momentum variables.

3. I realized that Dirac was working on this 1963 paper when I had an audience with him in 1962. He was talking about his paper at that time.

• Realizing that I was interested in squeezed states in optics, John Klauder sent me five copies of the paper he published with Yurke and MacCall in Phys. Rev. A. (1986), containing five of the ten generators given in Dirac's 1963 paper. After studying his paper, I started saying Paul A. M. Dirac invented squeezed states. Did you know? Don't forget Dirac's space-time squeeze as described before.

1. I met John Klauder when I went to Princeton in 1958. He had been Wheeler's student and received his degree in 1959. He has been very helpful to me especially whenever I needed his help. Many other friends of mine ran away from me when I needed them. You must have gone through the same experience if you lived long enough.

Here is a photo of myself talking to him in 1986. In this photo of 1994, John Klauder is negotiating with the residents of Nara's Deer Park in Japan.

2. During the period 1985-90, my job was to compose the stories Eugene Wigner wanted to hear. I was able to translate Dirac's 1963 on squeezed states into the language of Wigner functions.

3. In order to learn more about the Wigner function, I organized in 1986 a conference on the physics of phase space. Wigner came. Klauder came. So did many VIPs in the field, including Marlan Scully, Wolfgang Schleich, and John A. Wheeler. This conference served as the prelude to the international conference series now known as ICSSUR.

• Squeeze transformations play important roles also in understanding the two-by-two ABCD matrix. I enjoyed writing many papers on the ABCD matrices applicable to optical periodic systems including cavity optics, multilayer optics, as well as lens optics.

1. These optical processes can be described by the Lorentz group. More specifically, they contain the symmetry contents of Wigner's little groups dictating internal space-time symmetries in the Lorentz-covariant world.

2. Poincaré constructed his sphere in order to study polarization optics. Did you know that this sphere also contains the group theoretical contents of Dirac's 1963 paper on the squeezed state?

• Indeed, it was a rewarding experience for me to study the internal space-time symmetries of particles while writing papers in optical sciences.

 He built this castle in Osaka, and lived there.
1. Toyotomi Hideyosi is a very important person in Japanese history. In 1600, he unified his country, but Japan was too small for him. He thus appointed himself as the emperor of China. From his point of view at that time, China was the center of the world. He sent 150,000 Samurai troops to the Korean peninsula to make his way to go to Beijing for his coronation. The story could be very long, but let me stop here.

2. Toyotomi was such an unusual character that there are in Japan many stories about him. He apparently was not a handsome person.

Thus, he is a monkey to human eyes. He is a human if monkeys look at him. You have not seen this photo, but you have seen many photos of the Osaka Castle. He built this castle, and he lived there.

3. Likewise, I am an optics man to particle physicists, while I am a high-energy man to optical physicists.

• I seem to like Dirac. Click here to see how much I like him. You will be interested in the following pages.

 Dirac's bust in Princeton's Fine Hall Library. I talked with him in 1962 and in 1978.

2. Dirac and Feynman

3. Dirac as a poet

4. Dirac's Lorentz covariance

5. Dirac's Permutations

6. Dirac in Poland

7. and More

• I see rooms for squeezed in condensed matter physics. If you have ideas along this direction, let me know. Send me an email to me at yskim@umd.edu. We can work together. Squeeze transformations are everywhere in physics.