On the other hand, Yukawa claimed an unknown particle mediating "Strong Interaction" (strong force relating to nucleus).
The 2nd step of QED might tentatively be called here Latter Quantum Electrodynamics (QED) mostly completed by Feynman.
The non-existence of particles was rather realized, the term "Field" was employed representing a state of physical property like waves spread over Spacetime.
Then physics advances to other forces.
2.6.7.2 Details
2.6.7.2.1 Dirac Equation in 1928 CE
2.6.7.2.1.1 Background
Einstein presented the Theory of Relativity and revealed the errorness of Newtonian Mechanics.
Schrodinger presented Schrodinger Equation, but it based on Newtonian Mechanics excluding the Theory of Relativity. Then Schrodinger Equation had to be corrected following the Theory of Relativity. The first step was to fit the Special Theory of Relativity for now. First, Klein-Gordon Equation was presented.
It started from Mass-Energy Equivalence as follows.
E^2=m^2*c^4 + c^2*p^2 --------- Equation (7)
Then It was assumptively transformed following Quantum Mechanics compounding with ψ and changing "E" and "p" into operators as follows.
E(ope)^2 * ψ = m^2 * c^4 * ψ + c^2 * p(ope)^2 * ψ ----------- Equation (8)
Then substituting Equation (2) and (3) for Equation (8), Klein-Gordon Equation was derived. However, the Equation was quite complicated and seemed got stuck. In addition, it was unrelated to "spin (magnetic projection) quantum."
Looking back to Equation (10), E/c was defined as p0. Then Equation (12) would be rewritten as follows.
(γ0*E/c + γ1*p1 + γ2*p2 + γ3*p3 + m*c) * (γ0*E/c + γ1*p1 + γ2*p2 + γ3*p3 - m*c) = 0 ---- Equation (13)
If the latter term,
γ0*E/c + γ1*p1 + γ2*p2 + γ3*p3 - m*c = 0 --------------------- Equation (14),
E = - c * (γ1/γ0*p1 + γ2/γ0*p2 + γ3/γ0*p3) + m*c^2/γ0 -------- Equation (15)
Rewriting "E" into an operator, according to Equation (3),
E(ope) = i * hbar * ∂/∂t
= - c * (γ1/γ0*p1 + γ2/γ0*p2 + γ3/γ0*p3) + m*c^2/γ0 ---------- Equation (16)
1/γ0 means "I/γ0." ("I" is an identity matrix.)
On the other hand, γ0^2 = I
Then
1/γ0 = I/γ0 = γ0
Accordingly, Equation (16) is rewritten employing four-spinor ψ(t) as follows, where four-spinor ψ(t) here consists of 4 wave functions, ψ1, ψ2, ψ3, and ψ4.
Energy = E(ope) * ψ(t) = i * hbar * ∂/∂t * ψ(t)
= - c * (γ1*γ0*p1 + γ2*γ0*p2 + γ3*γ0*p3) * ψ(t) + m*c^2*γ0 * ψ(t) -------- Equation (17)
| Four-Spinor ψ(t) | ψ1 | ||
| ψ2 | |||
| ψ3 | |||
| ψ4 |
Assuming the particles stationary, momenta (p) are zero, Equation (17) changes as follows.
Energy = E(ope) * ψ(t) = i * hbar * ∂/∂t * ψ(t) = m*c^2*γ0 * ψ(t) -------- Equation (18)
Indicating matrices specifically,
| 1 | 0 | 0 | 0 | ψ1 | ||||||||||||
| 0 | 1 | 0 | 0 | ψ2 | ||||||||||||
| Energy = E(ope) * ψ(t) = i * hbar * ∂/∂t * ψ(t) = m*c^2 * | ||||||||||||||||
| 0 | 0 | -1 | 0 | ψ3 | ||||||||||||
| 0 | 0 | 0 | -1 | ψ4 |
Specifically, when the particles are stationary (momenta = zero) for example on ψ1,
i * hbar * ∂ψ1/∂t = m*c^2 * ψ1
| 1 | ||||||||||
| 0 | ||||||||||
| ψ1 = | e^(- i * m * c^2 * t/ hbar) | |||||||||
| 0 | ||||||||||
| 0 |
*Regarding Equation (13), the latter term Equation (14) (not the former term) was assumed zero. However, even if the former term is assumed zero
(γ0*E/c + γ1*p1 + γ2*p2 + γ3*p3 + m*c = 0) instead of the latter term (Equation (14)), the result is similar because γ0 symmetrically includes +1 and -1.
Then general Dirac Equation ranging over moving particles (momentum is not restricted to zero) through a complex mathematical procesure consequently for example on ψ1 is
| 1 | ||||||||||
| 0 | ||||||||||
| ψ1 = | e^(- i * (E+*t - p*x)/ hbar) | |||||||||
| Pz * c/(E+ + m * c^2) | ||||||||||
| (Px + i * py) * c/(E+ + m * c^2) |
2.6.7.2.1.2 Dirac's Claim
According to traditional algebra, Equation (7) wouldn't be clearly resolved. But, Dirac resolved Mass-Energy Equivalence (Equation (7)) into 2 parts employing matrices.
Equation (7) was transformed as follows.
(E/c)^2 - p^2 - (m*c)^2 = 0 -------- Equation (10)
E/c was defined as p0.
"p" was precisely extended as px, py, and pz and they were defined as p1, p2, and p3 corresponding to 3-dimentional space.
Then
p0^2 - p1^2 - p2^2 - p3^2 - (m*c)^2 = 0 -------- Equation (11)
Dirac transformed it as follows.
(γ0*p0 + γ1*p1 + γ2*p2 + γ3*p3 + m*c) * (γ0*p0 + γ1*p1 + γ2*p2 + γ3*p3 - m*c) = 0 -------- Equation (12)
According to traditional algebra, γ complying with the above equation would be absent. However, if γs are matrices, matrices complying with the above equation are present. Gamma Matrices (4 x 4 matrices) are the answer.
| γ0 |
γ1 |
γ2 |
γ3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -i | 0 | 0 | 1 | 0 | |||||||||||
| 0 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | i | 0 | 0 | 0 | 0 | -1 | |||||||||||
| 0 | 0 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | i | 0 | 0 | -1 | 0 | 0 | 0 | |||||||||||
| 0 | 0 | 0 | -1 | , | -1 | 0 | 0 | 0 | , | -i | 0 | 0 | 0 | , | 0 | 1 | 0 | 0 |
For example,
γ2*p2 * γ3*p3 + γ3*p3 * γ2*p2 = 0,
because
γ2 * γ3 + γ3 * γ2 = 0.
| 0 | 0 | 0 | -i | 0 | 0 | 1 | 0 | 0 | -i | 0 | 0 | ||||||||
| 0 | 0 | i | 0 | 0 | 0 | 0 | -1 | -i | 0 | 0 | 0 | ||||||||
| = | |||||||||||||||||||
| 0 | i | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -i | ||||||||
| -i | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -i | 0 | ||||||||
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | -i | 0 | i | 0 | 0 | ||||||||
| 0 | 0 | 0 | -1 | 0 | 0 | i | 0 | i | 0 | 0 | 0 | ||||||||
| = | |||||||||||||||||||
| -1 | 0 | 0 | 0 | 0 | i | 0 | 0 | 0 | 0 | 0 | i | ||||||||
| 0 | 1 | 0 | 0 | -i | 0 | 0 | 0 | 0 | 0 | i | 0 |
* "Gamma Matrices in Wikipedia" http://en.wikipedia.org/wiki/Gamma_matrices
*It should be noted that Gamma Matrices mostly include Pauli's Spin Matrices in the quarters.
* "Pauli Matrices in Wikipedia" http://en.wikipedia.org/wiki/Pauli_matrices
Dirac resolved Mass-Energy Equivalence through matrices and reached Dirac Equation. Dirac Equation corresponding to 4 x 4 matrices resulted in 4 equations and 4 wave functions. They were interpreted corresponding to 2 kinds of spins and 2 kinds of opposite particles including unknown particles.
Dirac claimed Dirac Equation complying with the Special Theory of Relativity, including "spin (magnetic projection) quantum number" of +1/2 and -1/2, it means spin (magnetic projection) quantum number of electrons. In addition, Dirac proposed the presence of opposite particles, "positron."
* "Dirac Equation in Wikipedia" http://en.wikipedia.org/wiki/Dirac_equation
Aside from that, Dirac proposed consecutive "creation and annihilation of particles" (later "virtual particles") everywhere in space introducing Creation and Annihilation Operators.
* "Positron in Wikipedia" http://en.wikipedia.org/wiki/Positron
* "Virtual Particle in Wikipedia" http://en.wikipedia.org/wiki/Virtual_particle
* "Creation and Annihilation Operators in Wikipedia" http://en.wikipedia.org/wiki/Creation_and_annihilation_operators
*By the way, Klein-Gordon Equation was consequently the equation related to "spin (magnetic projection) quantum number of 0," later "Higgs Particle" (Higgs Boson) proposed by Higgs in 1964 CE, discovered in 2012 CE.
* "Higgs Boson in Wikipedia" http://en.wikipedia.org/wiki/Higgs_boson
*On the other hand, Maxwell's Equations correspond to "spin (magnetic projection) quantum number of 1," it means spin (magnetic projection) quantum number of "light" and its particle "photon," proposed in Einstein's Light Quantum Hypothesis in 1905 CE.
2.6.7.2.2 Hubble's Law in 1929 CE
Background
Red Shift of the universe was reported.
Hubble's Law
Hubble examined relation between red shift and distance to the galaxies and confirmed red shift from galaxies is proportional to the distance to the galaxies. It supported the Big Bang Theory.
2.6.7.2.3 Weyl's Redisvovery of Vector Potential and U(1) Gauge Symmetric Theory in 1929 CE
2.6.7.2.3.1 Background
Physics had been seeking a new consistent theory to prospect unknown structure of the world. Maxwell's Equations were (are) commonly described employing E (electric field intensity) and B (magnetic flux density). However, Maxwell's Equations were originally described employing "Vector Potential" (A).
2.6.7.2.3.2 Weyl's Vector Potential and U(1) Gauge Symmetric Theory (or Gauge Theory).
On the way to attempting to unify gravity and electromagnetism, Weyl noted that employing Magnetic Vector Potential (commonly represented by A), instead of B (magnetic flux density), electromagnetic force provides clear U(1) symmetry (U(1) Gauge Invariance). Magnetic Vector Potential is constant (invariant) through U(1) transformations.
" B (magnetic flux density) = curl A (magnetic vector potential) " in mathematics. ("curl " is also described as "rot" derived from German or " ∇X ," representing "rotation" in a sense.)
* "Del in Wikipedia" http://en.wikipedia.org/wiki/Del
* "Magnetic Potential in Wikipedia" http://en.wikipedia.org/wiki/Magnetic_vector_potential#Magnetic_vector_potential
* "Introduction to Gauge Theory in Wikipedia" http://en.wikipedia.org/wiki/Introduction_to_gauge_theory
There could be some kinds of Gauge Symmetries, such as U(1) Gauge Symmetry, SU(2) Gauge Symmetry, and SU(3) Gauge Symmetry, consequently symmetries through rotation.
Yet Weyl's claim at that time focussed on U(1) Gauge Symmetry of electromagnetism (magnetic vector potential) for now. (SU(2) and SU(3) are matters of Yang-Mills Gauge Symmetry mentioned later.)
Other than that, it should be noted that such Gauge Symmetries are not associated with realistic space or spacetime. Gauge Symmetries are employed on abstract mathematical conceptual things such as Magnetic Vector Potential. Since Gauge Symmetries have nothing to do with realistic space or spacetime, they are called "internal symmetry" in "internal spaces." "Internal spaces" are unrealistic abstract mathematical conceptual things.
* "Symmetry (physics) in Wikipedia" http://en.wikipedia.org/wiki/Symmetry_(physics)
Unitary Transformations and Unitary Matrices
Unitary transformations are commonly operated through matrices. A Unitary Matrix or Unitary Matrices would be defined as " U† U =I ," where U is a Unitary matrix (naturally a square matrix), U† is "Conjugate Transpose Matrix" of U, " † " is "dagger," a sign indicating conjugate transpose matrix, " I " here is "Identity Matrix."
* "Unitary Matrix in Wikipedia" http://en.wikipedia.org/wiki/Unitary_matrix
* "Conjugate Transpose in Wikipedia" http://en.wikipedia.org/wiki/Conjugate_transpose
* "Identity Matrix in Wikipedia" http://en.m.wikipedia.org/wiki/Identity_matrix
As mentioned before, presuming the conjugate complex number and multiplying "the original complex number and the conjugate complex number" is the common way to calculate the size of the original complex number. On the other hand, according to the rules of Matrix Product, columns of the right (original) matrix correspond to rows of the left matrix. Then if the columns of the right matrix are changed into conjugate numbers and changed into the rows of the new left matrix, the original numbers of the columns could meet the brotherly conjugate numbers in the rows of the left new matrix through the multiplication. Transposing the columns to the rows results in adequate multiplication for each number.
This is the meaning of Conjugate Transpose Matrix. (Details are explained later in relation to Yang-Mills Theory) Yet it doesn't matter as far as U(1) transformation for now.
U(1) Transformation
U(1) matrices would be theoretically 1x1 unitary matrices, while virtually it is just multiplying a complex number with absolute value 1. It also means mere rotations in a (unrealistic conceptual) space consisting of a complex plane (a kind of Hilbert spaces).
(Hilbert spaces are, in a sense, spaces that axes of real numbers in ordinary spaces (Euclidean spaces) are replaced with axes of complex numbers (complex planes).)
For example, the following is an example of U(1) matrix.
| cos π/3 + i * sin π/3 | |||||
For reference, its conjugate transpose is as follows.
(Though, since this is 1x1 matrix, "transpose" has less meaning in this case.)
| cos π/3 - i * sin π/3 | |||||
Since the original matrix is a unitary matrix, multiplication of the conjugate transpose and the original matrix (1x1) is an identity matrix as follows.
| cos π/3 - i * sin π/3 | cos π/3 + i * sin π/3 | = | 1 | = | I | ||||||||||||||
Rotation
U(1) transformation results in a rotation as follows.
If " 4 " is multiplied by " cos π/3 + i * sin π/3 ," it results in " 4 * cos π/3 + 4 * i * sin π/3 " = " 2 + i * 3.464 ".
Then " 4 " is rotated on the complex plane as follows.
Since the distance from the center point is constant, such transformations would be called "Gauge Transformation."
Abelian Gauge Transformation
Aside from that, U(1) transformation is categorized as Abelian gauge transformation. As mentioned above, U(1) transformation is a rotation transformation.
Then for example supposing 2 kinds of U(1) operations, rotation by π/3 and rotation by π/4, the difference of orders of the 2 operations brings about the same results. For example, case (a): firstly rotate by π/3, secondly rotate by π/4 results in 7π/12 rotation. Similarly, case (b): firstly rotate by π/4, secondly rotate by π/3 results in 7π/12 rotation. Thus results are the same regardless of the orders of operations as far as U(1) transformation. Such transformations, including U(1) transformation, are called Abelian transformation.
U(1) Symmetry on Electromagnetic Force and Vector Potential's perfectness
Consequently, Electromagnetic Force related to electrons and photons has U(1) Symmetry through invariance (constancy) of Magnetic Vector Potential.
Other than that, according to Noether's Theorem, electric charge would be conserved along with this symmetry.
* "Symmetry Conservation Laws and Symmetry in Wikipedia" http://en.wikipedia.org/wiki/Symmetry_(physics)#Conservation_laws_and_symmetry
Thus Magnetic Vector Potential results in clear analysis. It also implies Magnetic Vector Potential's essential perfectness.
The essential perfectness of Magnetic Vector Potential is also implied by later Aharonov-Bohm Effect.
* "Aharonov-Bohm Effect in Wikipedia" http://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect
2.6.7.2.4 Pauli's Quantum Field Theory in 1930 CE
Background
Shrodinger and Dirac introduced the connection between electrons as particles and waves. However, details were uncertain.
Pauli's Field Theory
Pauli claimed non-existence of particles. Pauli claimed that the fundamental existence is properties of something (like waves) spread over space rather than particles.
* "Quantum Field Theory in Wikipedia" http://en.wikipedia.org/wiki/Quantum_field_theory
2.6.7.2.5 Anderson's Discovery of Positrons in 1932 CE
Background
Dirac predicted presence of positrons.
Anderson's Discovery of Positrons
Anderson discovered positrons.
(Presence of antiparticles and antimatter was realized.)
* "Antiparticle in Wikipedia" http://en.wikipedia.org/wiki/Antiparticle
* "Antimatter in Wikipedia" http://en.wikipedia.org/wiki/Antimatter
2.6.7.2.6 Chadwick's Discovery of Neutrons in 1932 CE
Background
Bothe found that beryllium exposed to alpha particles shows curious radiation.
Chadwick's Discovery of Neutrons
Chadwick claimed this new radiation is based on new particles with similar mass to protons but no electric charge, neutrons.
Subsequently, atomic nuclei were presumed consisting of protons and neutrons.
2.6.7.2.7 Heisenberg's Virtual Photon in 1932 CE
Background
Compton Scattering verified the property of particles (photons) about light.
Dirac claimed virtual particles.
Electromagnetic Force seemed associated with electrons, light, and photons. However, detailed mechanism was unclear.
* "Compton Scattering in Wikipedia" http://en.wikipedia.org/wiki/Compton_scattering
Heisenberg's Virtual Photon in 1932 CE
Heisenberg claimed Virtual Photons hover being created and annihilated around electrons and they mediate electromagnetic force.
2.6.7.2.8 Fermi's Beta Decay Theory and the Weak Force in 1934 CE
Background
Mechanism of Beta- Decay was suspected a conversion of a neutron to a proton and an electron, while some energy seemed lost and the energy conservation principle appeared violated, details were unclear.
Fermi's Claim
Fermi saw through Beta Decay that a neutron transforms into a proton, an electron, and a new particle embracing some energy (neutrino (later "antineutrino" to be precise)) presuming a new force, the Weak Force (Weak Interaction).
* "Beta Decay in Wikipedia" http://en.wikipedia.org/wiki/Beta_decay
* "Weak Interaction in Wikipedia" http://en.wikipedia.org/wiki/Weak_interaction
2.6.7.2.9 Einstein's EPR Paradox and Quantum Entanglement in 1935 CE
2.6.7.2.9.1 Background
Classical physics was based on the assumption like that a particle is like a point (dot). In contrast, Quantum Mechanics revealed that a particle is like a point (dot) in some cases but like a wave in other cases. Details of this wave-particle duality were unclear.
Then in relation to Quantum Mechanics and wave-particle duality, Bohr and Heisenberg claimed the Copenhagen Interpretation around 1925 CE. It claims as follows.
The original (initial) essential character of a particles is something like "existence probability" (Quantum Superposition) expressed as a wave function spread over a certain area rather than a realistic entity. However, once "something like existence probability spread over a certain area" is "observed (measured)," Wave Function Collapse occurs and the Quantum Superposition changes into a proper state (peculiar fundamental state). In other words, "observation (measurement)" changes the initial state "something like existence probability spread over a certain area" into a particle like a dot.
* "Copenhagen Interpretation in Wikipedia" http://en.wikipedia.org/wiki/Copenhagen_interpretation
* "Quantum Superposition in Wikipedia" http://en.wikipedia.org/wiki/Quantum_superposition
* "Wave Function Collapse in Wikipedia" http://en.wikipedia.org/wiki/Wave_function_collapse
2.6.7.2.9.2 EPR Paradox and Quantum Entanglement
Einstein opposed the Copenhagen Interpretation and advocated EPR Paradox (thought experiment) in 1935 CE, based on like Naive Realism and the Principle of Locality. Naive Realism would be like beliefs that human senses are quite reliable and for example tends to believe that things like material objects really exist in this world as humans perceive them. For example, Naive Realism might believe that green leaves exist. (However, they merely tend to reflect electromagnetic waves of 546 nm wavelength. Perceiving them to be green is a matter of humans' optical mechanism.) On the other hand, the Principle of Locality is an assumption that incidents, phenomena, or state, for example particles' change in a small area (local area) doesn't affect any distant area's state. In other words, the Principle of Locality believes like that an area's state is independent of (unrelated to) distant particles' behaviors.
EPR Paradox advocated like as follows. First, a wave (a particle) with (potential) spin 0 is assumed. (According to the Copenhagen Interpretation, it would be something like "existence probability" (Quantum Superposition) expressed as a wave function.) Second, the wave (particle) with potential spin 0 split into 2 waves (for example, an electron and a positron) wave-A (an electron) and wave-B (a positron), while wave-A and wave-B before observation are something like "existence probability" expressed as wave functions according to the Copenhagen Interpretation. Third, wave-A and wave-B drift somewhere respectively and consequently the locations of wave-A and wave-B become distant for example 1 light-year apart. (Observers are assumed, the assumed observer for A is named Alice and the assumed observer for B is named Bob.)
Attribution: http://en.wikipedia.org/wiki/File:EPR-paradox-illus.png
On the other hand, since wave-A and wave-B came from the original wave (original particle), the total property of wave-A and wave-B is equal to the original wave's (original particle's) property. For example, because of the Conservation of angular momentum, the total spin of particle-A and particle-B through observation should be 0 like the original particle (original wave). (Thus these 2 distant particles (waves) are related and called "entangled" or "entangled pair.")
The spins of wave-A and wave-B are unknown before observation. However, once the spin of A is observed by Alice and if it turned out to be "up spin," the spin of B is judged to be "down spin" without Bob's observation. It means the observed information was instantly transmitted to B 1 light-year distant contradicting to the Special Theory of Relativity, according to Einstein.
Based on like the above thought experiment (and assumed realism and locality), Einstein advocated incompleteness of Quantum Mechanics at that time and advocated another assumption. The assumption is that if Quantum Entanglement between distant waves (particles) is real, the both waves' (particles') observation results (up or down) should have been secretly fixed in advance prior to observation and the index of the results before observation is yet unknown or hidden. Then, if Quantum Entanglement actually occurs, the assumed hidden factor called "hidden variables" should be clarified and added to Quantum Mechanics.
* "EPR Paradox in Wikipedia" http://en.wikipedia.org/wiki/EPR_paradox
* "Naive Realism in Wikipedia" http://en.wikipedia.org/wiki/Na%C3%AFve_realism
* "Principle of Locality in Wikipedia" http://en.wikipedia.org/wiki/Principle_of_locality
2.6.7.2.10 Yukawa's Mesonic Theory in 1935 CE
Background
Nuclei were presumed consisting of protons and neutrons bindinding them. The mechanism of the binding was unclear. On the other hand, electromagnetic force was presumed mediated by photons.
Yukawa's Claim
Yukawa claimed presence of a new particle mediating the new force (Strong Force) binding between protons and neutrons. Yukawa predicted the mass of the new particle would be in between the proton/neutron and the electron, since short-range force would be mediated by particles with mass. The particles were later named "mesons" ("intermediate" in Greek), in contrast to "baryons" ("heavy" in Greek) such as protons and neutrons.
In addition, Yukawa claimed "forces" are mediated by particles. Long-range forces such as electromagnetic force would be mediated by particles with no mass. Short-range forces would be mediated by particles with mass.
* "Meson in Wikipedia" http://en.wikipedia.org/wiki/Meson
* "Baryon in Wikipedia" http://en.wikipedia.org/wiki/Baryon
2.6.7.2.11 Wigner's Isospin in 1937 CE
Background
Neutrons with similar mass to protons were discovered.
Wigner's Isospin
Wigner claimed protons and neutrons are basically the same and they are merely distinguished through a new quantum number. Wigner claimed the new quantum number Isospin to distinguish protons and neutrons. For example, a proton is numbered +1/2 of Isospin, a neutron is numbered -1/2 of Isospin.
* "Isospin in Wikipedia" http://en.wikipedia.org/wiki/Isospin
2.6.7.2.12 Weisskopf's Self-Energy in 1939 CE
Background
Positrons predicted by Dirac were discovered, Oppenheimer studied the fluctuations of positron-electron pairs in 1935 CE.
Weisskopf's Self-Energy
Weisskopf studied Self-Energy of electrons.
* "Self-Energy in Wikipedia" http://en.wikipedia.org/wiki/Self-energy
2.6.7.2.13 Dirac's Bra-Ket Notation in 1939 CE
Background
Schrodinger presented Wave Functions based on complicated mathematical analysis in an classical algebraic manner. However, classical algebraic analysis was rather complicated to advance to complex states.
Dirac's Bra-Ket Notation
Dirac advocated a versatile convenient way, Bra-Ket Notation, describing states of particles, quantum states, and so forth employing matrices or numerous numbers instead of classical algebraic functions.
* "Bra-Ket Notation in Wikipedia" http://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation
2.6.7.2.14 Fierz and Pauli's Spin-Statistics Theorem in 1940 CE
Background
Various particles were discovered and their properties were clarified.
Fierz and Pauli's Spin-Statistics Theorem
Fierz and Pauli claimed particles are classified according to spin quantum numbers. A particle has either integer or half-integer spin quantum number.
Particles with integer spin quantum number are classified as Bosons.
Particles with half-integer spin quantum number are classified as Fermions.
If positions of any 2 particles with the same integer spin quantum numbers were swapped, the wavefunctions of the system on Bosons have the same values (wavefunctions of Bosons are symmetric). Bosons such as photons correspond to nonmaterial entities.
In contrast, if positions of any 2 particles with the same half-integer spin quantum numbers were swapped, the wavefunctions of the system on Fermions have different values (the sign is changed) (wavefunctions of Fermions are anti-symmetric). Fermions such as electrons correspond to particles of "substances (matter)."
*Particles primarily associated with forces are Bosons. The Gauge Symmetric Theory mentioned above is a theory relating to forces and Bosons.
2.6.7.2.15 Discovery of Mesons in 1947 CE
Background
Yukawa claimed presence of Mesons.
Discovery of Mesons
Powell discovered π mesons in 1947 CE. The discovered π mesons at that time were the π+ meson and the π- meson to be precise.
(There are 3 kinds of π mesons including the π0 meson discovered later. These 3 mesons are media of the Strong Force.)
Rochester and Butler discovered K mesons in 1947 CE. K mesons had a strange property.
2.6.7.2.16 Bethe's Improvement of Vacuum Polarization and QED Vacuum in 1947 CE
Background
Dirac claimed consecutive creation and annihilation of particles (virtual particles) everywhere in space.
The Uncertainty Principle was realized.
Bethe's Improvement of Vacuum Polarization and QED Vacuum
Bethe calculated Vacuum Polarization.
Then consecutive creation and annihilation of particles everywhere in space (including vacuum space) on electrons were theorized based on Quantum Fluctuation came from the Uncertainty Principle.
Quantum Fluctuation is temporary change in reference to time and energy (equivalent to mass or particles). In other words, particles may emerge and disappear in a short time. It was accepted through the Uncertainty Principle.
* "Vacuum Polarization in Wikipedia" http://en.wikipedia.org/wiki/Vacuum_polarization
* "QED Vacuum in Wikipedia" http://en.wikipedia.org/wiki/QED_vacuum
* "Quantum Fluctuation in Wikipedia" http://en.wikipedia.org/wiki/Quantum_fluctuation
* "Nelson Scientific" http://home.comcast.net/~nelsonscientific/scientific_discussion.html
* "Casimir Effect in Wikipedia" http://en.wikipedia.org/wiki/Casimir_effect
*Consequently, affairs around an electron would be illustrated as follows.
Virtual Photons and Virtual Electron-Positron Pairs are created and annihilated around an electron.
If there are other electrons, positrons, or protons close to them, the virtual photons would mediate Electromagnetic Force.
Other than that, if the electron leaves quickly, the virtual photons are remained losing their base. Then the virtual photons are radiated. However, if an electron appears and disappears repeatedly, virtual photons are created and radiated repeatedly. Then the fluctuation of radiated photons induces electrons, fluctuation of electrons, and electromagnetic waves (light).
2.6.7.2.17 Schwinger, Feynman and Tomonaga's Renormalization in 1948 CE
Background
Specific calculation method (approximation) on Quantum Mechanics has been Perturbation Theory Calculation. However, Perturbation Calculation at that time held mathematical difficulty for example on Electron Self-Energy, where an electron emits a photon and absorbs the emitted photon.
Schwinger, Feynman and Tomonaga's Renormalization
Schwinger, Feynman and Tomonaga claimed modification of theoretical contradiction by Renormalization Theory.
* "Electron Self-Energy in Wikimedia" http://commons.wikimedia.org/wiki/File:Electron_self_energy_loop.svg
* "Perturbation Theory in Wikipedia" http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)
* "Renormalization in Wikipedia" http://en.wikipedia.org/wiki/Renormalization
2.6.7.2.18 Feynman's Path Integral and Feynman Diagram in 1948 CE
Background
According to the traditional physics, elements of the world or substances were presumed to be particles.
On the other hand, de Broglie presented the concept of matter wave and Schrodinger, to a certain extent, specified the matter wave in relation to a kind of matter, electrons. However, the relation between particles and waves was unclear.
On the other hand, basic concept of integrating paths was presented.
Feynman's Path Integral
Feynman presented specific method of Path Integral.
For example, a trajectory of a particle (or a substance) based on the traditional particle physics (or Newtonian Mechanics) would be depicted as follows.
On the other hand, transmission of waves would be depicted as follows. (Oranges correspond to the tops of waves and blues correspond to bottoms of waves.) Directions or paths of waves' transmission are at random and diverse as depictions of the Huygens Principle suggest. Paths of waves widely and randomly spread.
* "Yugawad Huygens Principle" http://yugawad.files.wordpress.com/2012/04/huygens_principle.gif
* "Huygens Fresnel Principle in Wikipedia" http://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle
The question is how these 2 concepts relate.
The solution lies in "Extinction of Waves through Interference of Waves."
If shapes of 2 paths of waves are similar (oranges of the 2 paths meet and blues of the 2 paths meet), the wave would be amplified through interference. However, if shapes of 2 paths of waves differ (oranges don't meet and blues don't meet), the waves would be extinguished through interference and this is usually the case.
Then despite widely spread paths of waves, most paths of waves are extinguished and one path (or a few paths) remains. The remained path of waves corresponds to the trajectory of a particle by the traditional particle physics. In other words, integration of all paths results in one path corresponding to the trajectory of a particle by the traditional particle physics.
From a viewpoint of Least Action on the paths of waves, Action (S) increaes where waves are extinguished or weakened through interference, while Action (S) decreases where waves are amplified through interference. Then the remained path corresponds to the path of Least Action on waves. The path of waves of Least Action naturally corresponds to the trajectory of a particle or a substance (matter), since a substance (matter) of Newtonian mechanics is an aggregation of elementary particles, in other words, an aggregation of remained waves.
In other words, a particle is merely an aspect of waves or a simplified expression of waves. Extinction of waves through interference caused misunderstanding assuming presence of particles. The essential existence is waves rather than particles.
In addition, since Newtonian mechanics is aggregations of remained waves, Lagrangian of Newtonian mechanics (T-V) is aggregations of Lagrangian on remained waves. (The origin of Lagrangian of Newtonian mechanics lies in Lagrangian on waves (or elementary particles).)
This is the meaning of Path Integral.
* "Path Integral Formulation in Wikipedia" http://en.wikipedia.org/wiki/Path_integral_formulation
*The non-existence of particles was rather realized, the term "Field" (for example "electron field" and "photon field") was employed representing a state of property like waves associated with the concept of elementary particles spread over Spacetime.
* "Quantum Field in Wikipedia" http://en.wikipedia.org/wiki/Quantum_field
* "Quantum Field Theory in Wikipedia" http://en.wikipedia.org/wiki/Quantum_field_theory
*Particles are merely remained (amplified) parts of waves. The essential entities of particles are waves spread over certain areas. The Uncertainty Principle, virtual particles, and particles mediating forces should be interpreted in this sense.
Since particles are merely remained (amplified) parts of waves spread over certain areas, locations of particles (remained amplified part) change depending on change of waves' states (Uncertainty Principle).
Since particles are merely remained (amplified) parts of waves, particles (remained amplified parts) would emerge and annihilate temporarily (virtual particles).
Since the essential entities of particles are waves spread over certain areas, particles (remained parts of waves spread over certain areas) could mediate (affect) motion of other particles (states of other waves) (particles' mediation of forces).
*However, the term "particle" still continues to be employed later as a convenient term representing the remained part of waves.
In addition, Feynman presented Feynman Diagram, a simple way describing interaction of particles.
* "Feynman Diagram in Wikipedia" http://en.wikipedia.org/wiki/Feynman_diagram
Thus Quantum Electrodynamics was largely completed.