2. The Action Quantization Method in Quantum Mechanics
After discovering this new action for classical electrodynamics, Feynman attempted to quantize it in hopes of obtaining a satisfactory theory of quantum electrodynamics. At that time, the action method had not yet been adopted in quantum physics. The conventional route was to start from the Hamiltonian function, replace the position and momentum in the classical Hamiltonian with operators, and then apply non-commutation relations. Feynman did not know at the time that Dirac had already introduced the action and the Lagrangian function into quantum mechanics in a 1932 article [9]. While he was struggling for an answer, a European scholar visiting Princeton told him that Dirac had discussed this problem in a certain article. Upon hearing this information, Feynman went to the library the next day to read the paper.
In his 1932 article, Dirac introduced a very important function \langle q_{t+dt} | q_t \rangle and pointed out that it "corresponds to" \exp[\frac{i}{\hbar}Ldt] [9]. This "means," Dirac emphasized: "We should not consider the classical Lagrangian as a function of coordinates and velocities, but rather as a function of coordinates at two different times t and t+dt." [9] Inspired by Dirac’s thought, Feynman directly changed "corresponds to" to "proportional to":
\Psi(x', t+\varepsilon) = \int K(x', t+\varepsilon; x, t) \Psi(x, t) dx = \int A \exp\left[ \frac{i\varepsilon}{\hbar} L\left( \frac{x'-x}{\varepsilon}, x \right) \right] \Psi(x, t) dx
And for the specific case of L = \frac{1}{2} m\dot{x}^2 - V(x), he derived the Schrödinger equation from the above expression. Based on this, Feynman completed his doctoral dissertation, "The Principle of Least Action in Quantum Mechanics," in 1942. In this paper, he derived a brand-new concept—a "space-time" description of probability amplitudes that satisfy the superposition principle. The traditional wave function now became the probability amplitude from an initial state to a final state. Based on this idea, Feynman questioned the hypothesis of wave packet reduction.
In 1942, the Pacific War broke out, and Feynman’s theoretical exploration was consequently delayed. In fact, even before completing his doctoral dissertation, Feynman had already joined the Manhattan Project. In the spring of 1943, he left Princeton for Los Alamos to lead the computing group of the theoretical division of the project. At Los Alamos, his candid personality, sharp insight, and outstanding performance won high praise from older generation physicists such as Bohr and Bethe; Wigner hailed him as a second Dirac. In the spring of 1946, the Manhattan Project was completed, and Feynman moved to the Department of Physics at Cornell University. By then, his first wife, Arline, had already passed away in New Mexico. However, heavy work and life’s tragedies did not stifle Feynman’s quest for the "rules of the game" [10]. Even at Los Alamos, he always carried small slips of paper, often lost in thought on buses or in the streets, trying to quantize the action he had discovered. At this time, he encountered two major difficulties: one involved spin-1/2 relativistic electrons, and the other was the problem of photons, which is the second term of the action A.
Regarding the electron, Feynman’s goal was to establish a simpler relativistic electron theory within the framework of complex probability amplitudes for paths, rather than following the Dirac equation. He believed that the simplicity of nature allows us to view the same thing in different ways. Therefore, he dreamed that the Dirac four-component spinor, Dirac matrices, particle pair creation and annihilation, the negative energy sea, etc., could all be derived from his new theory. In the one-dimensional case, Feynman found great promise. Assuming an electron moves back and forth at the speed of light and dividing time into intervals of length \varepsilon, where the electron only changes its direction of velocity at the endpoints of each small interval, then the total probability amplitude of a path is (i\varepsilon)^N, where N is the number of direction changes. From one space-time point to an adjacent space-time point, there are two paths (left and right), and the total probability amplitude is the sum of the amplitudes of the two paths, one of which has an extra factor of i\varepsilon compared to the other. Letting \varepsilon \to 0, Feynman derived the two-component Dirac equation in two-dimensional space-time. Feynman then tried to generalize it to three-dimensional space, but all efforts ended in failure. Finally, he had to temporarily return to the non-relativistic case, taking the first term of the action A as \frac{1}{2} m\dot{x}^2.
Regarding the second term of the action A, Feynman initially thought there was no problem because he could use the new action to define energy and momentum, and he also proved that as long as special boundary conditions satisfying half-retarded and half-advanced solutions were added, his action was equivalent to the Frenkel field (ignoring the field produced by the electron under consideration). However, to describe the traditional retarded interaction theory within his framework, he had to write out the probability amplitude, which meant performing double path integrals. To this end, Feynman conducted extensive calculations and verifications for various boundary conditions and various forms of action: for example, using f(s^2) or s_+ instead of \delta. But the results were unsatisfactory because the energy he defined now became a complex number, and the probability could not maintain unitarity.
Five years of hard thinking did not achieve his goal: a satisfactory theory of quantum electrodynamics. But it was precisely through this arduous reflection that Feynman obtained a "wonderful world-view, woven from world-lines in space-time, where everything can move as it pleases, and what actually happens is the sum of all possible histories." ([1], p.61) Returning to the non-relativistic case, based on his doctoral dissertation, he derived the third formulation of quantum mechanics—the path integral method of action quantization. In 1948, he published this startling theory, "Space-Time Approach to Non-Relativistic Quantum Mechanics," in Reviews of Modern Physics. Here, Feynman started from the superposition principle of probability amplitudes and completely established the path integral theory using the action quantization method. The total probability amplitude from one space-time point to another is the sum of the probability amplitudes of all possible paths, and the probability amplitude of each path is \exp[\frac{i}{\hbar}S], where S corresponds to the classical action of that path.
By introducing action into quantum mechanics, Feynman built a new bridge connecting classical mechanics and quantum mechanics. In Feynman’s theory, the meaning of the Planck constant \hbar is distinct. R. Hund, author of The History of Quantum Theory, once said, "Quantum theory is the study of the role of h in nature." [11]. In Planck’s theory, h has no independent meaning; it is linked with frequency, meaning h\nu is the energy quantum. According to the Bohr-Sommerfeld theory, electrons in an atom can only move around the nucleus in certain specific possible orbits, which are determined by the angular momentum quantization condition \oint p dq = n\hbar, where \hbar is the basic unit of angular momentum. In matrix mechanics, \hbar appears in non-commutation relations, and its meaning is the statistical dispersion of conjugate mechanical quantities. In wave mechanics, \hbar is the protagonist linking wave-particle duality, P \to -i\hbar\frac{\partial}{\partial x}, E \to i\hbar\frac{\partial}{\partial t}. Now, according to Feynman’s theory, \hbar appears in the phase of the probability amplitude, so it must have the same dimensions as action [12]. In other words, it is the quantum of action.
Using the action quantization method, the concept of probability amplitude emerged as the most fundamental concept in quantum mechanics. After the 1960s, Feynman’s views exerted a huge influence on the younger generation of physicists through his lectures and writings. Regarding this, S.S. Schweber commented: "His doctoral dissertation, which elucidated the path integral form of non-relativistic quantum mechanics, and his 1947 article in Reviews of Modern Physics, allow us to grasp the basic assumptions of standard quantum mechanics describing microscopic entities with extreme clarity. His formulation is a brand-new form that transcends conventional formulations. It can be said that his reformulation of quantum mechanics and his ’path integral’ will be his most profound and lasting contributions to physics. They not only greatly deepened our understanding of quantum mechanics but also greatly expanded the systems that can be quantized." [13]
By the 1970s, the older generation of physicists had changed their views on the core concepts of quantum mechanics. In 1972, Heisenberg said in a speech celebrating Dirac’s 70th birthday: "The concept of state... can only be understood after the establishment of quantum mechanics and wave mechanics." Indeed, the so-called wave function should correctly be called a state function. If we consider retarded interactions, the form of the Lagrangian is L(X(t), X(t+T)), which is a function of positions at two different times. At this point, applying the Hamiltonian method would be very clumsy; applying the path integral method, we do not need to specify the details of every moment, nor do we need to use differential equations to derive \Psi(x, t) from \Psi(x, t'), but only directly consider all possible paths from \Psi(x_a, t_a) to \Psi(x_b, t_b). In this way, the wave function loses its meaning, and wave-particle duality and the complementarity principle now become the scaffolding for building the edifice of quantum mechanics. The uncertainty relations in quantum mechanics, as Lindsay and Margenau stated, directly stem from the definition of state unique to quantum mechanics.
The concept of state in quantum mechanics was brilliantly analyzed by Mr. Guan Hong in his book Basic Concepts in Quantum Mechanics ([4], Chapter 6). Guan Hong pointed out that the state function itself is not a dynamical variable, but the eigenvalues of a set of dynamical variables determine a specific quantum mechanical state. How many mechanical variables are needed to describe a physical system depends on our knowledge of that system. With the progress of particle physics, we have successively added new mechanical variables to a particle, such as spin, isospin, parity, strangeness, and color charge. The specific form of a mechanical variable must withstand the test of physical experiments; the correspondence principle is only a heuristic tool, not the core of quantum theory. Take the Hamiltonian, for example: \frac{P^2}{2m} has undergone extensive testing, the Coulomb potential \frac{Ze^2}{r} was confirmed by atomic energy level splitting and scattering experiments, and the Newtonian potential \frac{GMm}{r} was not determined until the neutron interference experiments of the 1970s. In quantum theory, a mechanical variable must take the form of an operator. But not all operators can be written as expressions of position and momentum, such as spin and isospin; therefore, non-commutation relations are only specific regulations for certain operators. In the 1960s, Dirac still regarded non-commutation relations as the basic feature of quantum mechanics. By the 1970s, Dirac also turned to believe that the basic feature of quantum mechanics is the existence of probability amplitudes, rather than his beloved non-commutative algebra, and said that it was precisely through the genius of Heisenberg and Schrödinger that the existence of probability amplitudes with phases was revealed to people [16].
"The concept of probability in quantum mechanics has not changed." In the book Quantum Mechanics and Path Integrals, Feynman pointed out from the very beginning: "What has changed, and fundamentally changed, is the method of calculating probabilities." [17] Obviously, Feynman’s view is consistent with the spirit of statistical interpretation, but he did not break with the Copenhagen view. "Probability amplitudes are almost inconceivable" ([8], p.166), Feynman said, yet "so far no one has seen through its essence."
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