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93-417: The spin magnetic moment of a charged, spin-1/2 particle that does not possess any internal structure (a Dirac particle) is given by μ = g e 2 m S , {\displaystyle {\boldsymbol {\mu }}=g{e \over 2m}\mathbf {S} ,} where μ is the spin magnetic moment of the particle, g is the g -factor of the particle, e is the elementary charge , m

186-438: A positron moving forward in time.) Quantum mechanics introduces an important change in the way probabilities are computed. Probabilities are still represented by the usual real numbers we use for probabilities in our everyday world, but probabilities are computed as the square modulus of probability amplitudes , which are complex numbers . Feynman avoids exposing the reader to the mathematics of complex numbers by using

279-465: A Feynman diagram could be drawn describing it. This implies a complex computation for the resulting probability amplitudes, but provided it is the case that the more complicated the diagram, the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of any interactive process between electrons and photons, it

372-439: A better estimation for the total probability amplitude by adding the probability amplitudes of these two possibilities to our original simple estimate. Incidentally, the name given to this process of a photon interacting with an electron in this way is Compton scattering . There is an infinite number of other intermediate "virtual" processes in which more and more photons are absorbed and/or emitted. For each of these processes,

465-511: A complete account of matter and light interaction. In technical terms, QED can be described as a very accurate way to calculate the probability of the position and movement of particles, even those massless such as photons, and the quantity depending on position (field) of those particles, and described light and matter beyond the wave-particle duality proposed by Albert Einstein in 1905. Richard Feynman called it "the jewel of physics" for its extremely accurate predictions of quantities like

558-452: A finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named renormalization . Based on Bethe's intuition and fundamental papers on the subject by Shin'ichirō Tomonaga , Julian Schwinger , Richard Feynman and Freeman Dyson , it was finally possible to get fully covariant formulations that were finite at any order in

651-442: A first order of perturbation theory , a problem already pointed out by Robert Oppenheimer . At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics . Difficulties with

744-498: A later time) and a photon at D (yet another place and time)?". The simplest process to achieve this end is for the electron to move from A to C (an elementary action) and for the photon to move from B to D (another elementary action). From a knowledge of the probability amplitudes of each of these sub-processes – E ( A to C ) and P ( B to D ) – we would expect to calculate the probability amplitude of both happening together by multiplying them, using rule b) above. This gives

837-436: A line, it breaks up into a collection of "simple" lines, each of which, if looked at closely, are in turn composed of "simple" lines, and so on ad infinitum . This is a challenging situation to handle. If adding that detail only altered things slightly, then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to infinite probability amplitudes. In time this problem

930-558: A perturbation series of quantum electrodynamics. Shin'ichirō Tomonaga, Julian Schwinger and Richard Feynman were jointly awarded with the 1965 Nobel Prize in Physics for their work in this area. Their contributions, and those of Freeman Dyson , were about covariant and gauge-invariant formulations of quantum electrodynamics that allow computations of observables at any order of perturbation theory . Feynman's mathematical technique, based on his diagrams , initially seemed very different from

1023-708: A result of the mass difference between the particles. However, not all of the difference between the g -factors for electrons and muons is exactly explained by the Standard Model . The muon g -factor can, in theory, be affected by physics beyond the Standard Model , so it has been measured very precisely, in particular at the Brookhaven National Laboratory . In the E821 collaboration final report in November 2006,

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1116-400: A set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an evolution operator , which for a given initial state | i ⟩ {\displaystyle |i\rangle } will give a final state ⟨ f | {\displaystyle \langle f|} in such

1209-436: A simple but accurate representation of them as arrows on a piece of paper or screen. (These must not be confused with the arrows of Feynman diagrams, which are simplified representations in two dimensions of a relationship between points in three dimensions of space and one of time.) The amplitude arrows are fundamental to the description of the world given by quantum theory. They are related to our everyday ideas of probability by

1302-452: A simple estimated overall probability amplitude, which is squared to give an estimated probability. But there are other ways in which the result could come about. The electron might move to a place and time E , where it absorbs the photon; then move on before emitting another photon at F ; then move on to C , where it is detected, while the new photon moves on to D . The probability of this complex process can again be calculated by knowing

1395-515: Is K J = 2 e h , {\displaystyle K_{\text{J}}={\frac {2e}{h}},} where h is the Planck constant . It can be measured directly using the Josephson effect . The von Klitzing constant is R K = h e 2 . {\displaystyle R_{\text{K}}={\frac {h}{e^{2}}}.} It can be measured directly using

1488-413: Is quantum chromodynamics , which began in the early 1960s and attained its present form in the 1970s work by H. David Politzer , Sidney Coleman , David Gross and Frank Wilczek . Building on the pioneering work of Schwinger , Gerald Guralnik , Dick Hagen , and Tom Kibble , Peter Higgs , Jeffrey Goldstone , and others, Sheldon Glashow , Steven Weinberg and Abdus Salam independently showed how

1581-402: Is shot noise . Shot noise exists because a current is not a smooth continual flow; instead, a current is made up of discrete electrons that pass by one at a time. By carefully analyzing the noise of a current, the charge of an electron can be calculated. This method, first proposed by Walter H. Schottky , can determine a value of e of which the accuracy is limited to a few percent. However, it

1674-400: Is a constant, and is related to, but not the same as, the measured electron charge e . QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that

1767-501: Is a fundamental physical constant , defined as the electric charge carried by a single proton (+ 1e) or, equivalently, the magnitude of the negative electric charge carried by a single electron , which has charge −1  e . In the SI system of units , the value of the elementary charge is exactly defined as e {\displaystyle e} = 1.602 176 634 × 10 coulombs , or 160.2176634 zepto coulombs (zC). Since

1860-465: Is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability amplitude. That basic scaffolding remains when one moves to a quantum description, but some conceptual changes are needed. One is that whereas we might expect in our everyday life that there would be some constraints on

1953-413: Is a one-to-one correspondence between the electrons passing through the anode-to-cathode wire and the ions that plate onto or off of the anode or cathode. Measuring the mass change of the anode or cathode, and the total charge passing through the wire (which can be measured as the time-integral of electric current ), and also taking into account the molar mass of the ions, one can deduce F . The limit to

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2046-787: Is also credited with coining the term "quantum electrodynamics". Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli , Eugene Wigner , Pascual Jordan , Werner Heisenberg and an elegant formulation of quantum electrodynamics by Enrico Fermi , physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck , and Victor Weisskopf , in 1937 and 1939, revealed that such computations were reliable only at

2139-897: Is an abelian gauge theory with the symmetry group U(1) , defined on Minkowski space (flat spacetime). The gauge field , which mediates the interaction between the charged spin-1/2 fields , is the electromagnetic field . The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action S QED = ∫ d 4 x [ − 1 4 F μ ν F μ ν + ψ ¯ ( i γ μ D μ − m ) ψ ] {\displaystyle S_{\text{QED}}=\int d^{4}x\,\left[-{\frac {1}{4}}F^{\mu \nu }F_{\mu \nu }+{\bar {\psi }}\,(i\gamma ^{\mu }D_{\mu }-m)\,\psi \right]} where Expanding

2232-421: Is as follows: where a shorthand symbol such as x A {\displaystyle x_{A}} stands for the four real numbers that give the time and position in three dimensions of the point labeled A . A problem arose historically which held up progress for twenty years: although we start with the assumption of three basic "simple" actions, the rules of the game say that if we want to calculate

2325-416: Is defined; see below.) This is the reason for the terminology "elementary charge": it is meant to imply that it is an indivisible unit of charge. There are two known sorts of exceptions to the indivisibility of the elementary charge: quarks and quasiparticles . All known elementary particles , including quarks, have charges that are integer multiples of ⁠ 1 / 3 ⁠   e . Therefore,

2418-570: Is exactly defined since 20 May 2019 by the International System of Units . Prior to this change, the elementary charge was a measured quantity whose magnitude was determined experimentally. This section summarizes these historical experimental measurements. If the Avogadro constant N A and the Faraday constant F are independently known, the value of the elementary charge can be deduced using

2511-692: Is its spin angular momentum, and μ B = eħ /2 m e is the Bohr magneton . In atomic physics, the electron spin g -factor is often defined as the absolute value of g e : g s = | g e | = − g e . {\displaystyle g_{\text{s}}=|g_{\text{e}}|=-g_{\text{e}}.} The z -component of the magnetic moment then becomes μ z = − g s μ B m s {\displaystyle \mu _{\text{z}}=-g_{\text{s}}\mu _{\text{B}}m_{\text{s}}} The value g s

2604-641: Is its orbital angular momentum, and μ B is the Bohr magneton. For an infinite-mass nucleus, the value of g L is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical magnetogyric ratio . For an electron in an orbital with a magnetic quantum number m l , the z -component of the orbital magnetic moment is μ z = − g L μ B m l {\displaystyle \mu _{\text{z}}=-g_{L}\mu _{\text{B}}m_{\text{l}}} which, since g L = 1,

2697-496: Is roughly equal to 2.002319 and is known to extraordinary precision – one part in 10. The reason it is not precisely two is explained by quantum electrodynamics calculation of the anomalous magnetic dipole moment . The spin g -factor is related to spin frequency for a free electron in a magnetic field of a cyclotron: ν s = g 2 ν c {\displaystyle \nu _{\text{s}}={\frac {g}{2}}\nu _{\text{c}}} Secondly,

2790-436: Is the Planck constant , α is the fine-structure constant , μ 0 is the magnetic constant , ε 0 is the electric constant , and c is the speed of light . Presently this equation reflects a relation between ε 0 and α , while all others are fixed values. Thus the relative standard uncertainties of both will be same. Quantum electrodynamics In particle physics , quantum electrodynamics ( QED )

2883-421: Is the electron spin g-factor (more often called simply the electron g-factor ), g e , defined by μ s = g e μ B ℏ S {\displaystyle {\boldsymbol {\mu }}_{\text{s}}=g_{\text{e}}{\mu _{\text{B}} \over \hbar }\mathbf {S} } where μ s is the magnetic moment resulting from the spin of an electron, S

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2976-454: Is the relativistic quantum field theory of electrodynamics . In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving

3069-473: Is the speed of light , ε 0 is the electric constant , and ħ is the reduced Planck constant . Charge quantization is the principle that the charge of any object is an integer multiple of the elementary charge. Thus, an object's charge can be exactly 0  e , or exactly 1  e , −1  e , 2  e , etc., but not ⁠ 1 / 2 ⁠   e , or −3.8  e , etc. (There may be exceptions to this statement, depending on how "object"

3162-575: Is the magnetic moment of the nucleon or nucleus resulting from its spin, g is the effective g -factor, I is its spin angular momentum, μ N is the nuclear magneton, e is the elementary charge, and m p is the proton rest mass. There are three magnetic moments associated with an electron: one from its spin angular momentum , one from its orbital angular momentum , and one from its total angular momentum (the quantum-mechanical sum of those two components). Corresponding to these three moments are three different g -factors: The most known of these

3255-441: Is the mass of the particle, and S is the spin angular momentum of the particle (with magnitude ħ /2 for Dirac particles). Protons, neutrons, nuclei, and other composite baryonic particles have magnetic moments arising from their spin (both the spin and magnetic moment may be zero, in which case the g -factor is undefined). Conventionally, the associated g -factors are defined using the nuclear magneton, and thus implicitly using

3348-421: Is the total magnetic moment resulting from both spin and orbital angular momentum of an electron, J = L + S is its total angular momentum, and μ B is the Bohr magneton . The value of g J is related to g L and g s by a quantum-mechanical argument; see the article Landé g -factor. μ J and J vectors are not collinear, so only their magnitudes can be compared. The muon, like

3441-401: Is to say that their orientations in space and time have to be taken into account. Therefore, P ( A to B ) consists of 16 complex numbers, or probability amplitude arrows. There are also some minor changes to do with the quantity j , which may have to be rotated by a multiple of 90° for some polarizations, which is only of interest for the detailed bookkeeping. Associated with the fact that

3534-435: Is unambiguous: it refers to a quantity of charge equal to that of a proton. Paul Dirac argued in 1931 that if magnetic monopoles exist, then electric charge must be quantized; however, it is unknown whether magnetic monopoles actually exist. It is currently unknown why isolatable particles are restricted to integer charges; much of the string theory landscape appears to admit fractional charges. The elementary charge

3627-415: Is very important: it means that there is no observable feature present in the given system that in any way "reveals" which alternative is taken. In such a case, one cannot observe which alternative actually takes place without changing the experimental setup in some way (e.g. by introducing a new apparatus into the system). Whenever one is able to observe which alternative takes place, one always finds that

3720-543: Is written Expanding the covariant derivative in the Lagrangian gives For simplicity, B μ {\displaystyle B_{\mu }} has been set to zero. Alternatively, we can absorb B μ {\displaystyle B_{\mu }} into a new gauge field A μ ′ = A μ + B μ {\displaystyle A'_{\mu }=A_{\mu }+B_{\mu }} and relabel

3813-656: Is − μ B m l For a finite-mass nucleus, there is an effective g value g L = 1 − 1 M {\displaystyle g_{L}=1-{\frac {1}{M}}} where M is the ratio of the nuclear mass to the electron mass. Thirdly, the Landé g-factor , g J , is defined by | μ J | = g J μ B ℏ | J | {\displaystyle |{\boldsymbol {\mu _{\text{J}}}}|=g_{J}{\mu _{\text{B}} \over \hbar }|\mathbf {J} |} where μ J

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3906-590: The U ( 1 ) {\displaystyle {\text{U}}(1)} current j μ {\displaystyle j^{\mu }} as ∂ μ F μ ν = e j ν . {\displaystyle \partial _{\mu }F^{\mu \nu }=ej^{\nu }.} Now, if we impose the Lorenz gauge condition ∂ μ A μ = 0 , {\displaystyle \partial _{\mu }A^{\mu }=0,}

3999-444: The 2019 revision of the SI , the seven SI base units are defined in terms of seven fundamental physical constants, of which the elementary charge is one. In the centimetre–gram–second system of units (CGS), the corresponding quantity is 4.803 2047 ... × 10   statcoulombs . Robert A. Millikan and Harvey Fletcher 's oil drop experiment first directly measured the magnitude of

4092-528: The Fermilab Muon g −2 collaboration presented and published a new measurement of the muon magnetic anomaly. When the Brookhaven and Fermilab measurements are combined, the new world average differs from the theory prediction by 4.2 standard deviations. The electron g -factor is one of the most precisely measured values in physics. Elementary charge The elementary charge , usually denoted by e ,

4185-479: The Shelter Island Conference . While he was traveling by train from the conference to Schenectady he made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford . Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and charge that were actually fixed to

4278-444: The anomalous magnetic dipole moment . However, as Feynman points out, it fails to explain why particles such as the electron have the masses they do. "There is no theory that adequately explains these numbers. We use the numbers in all our theories, but we don't understand them – what they are, or where they come from. I believe that from a fundamental point of view, this is a very interesting and serious problem." Mathematically, QED

4371-463: The anomalous magnetic moment of the electron and the Lamb shift of the energy levels of hydrogen . It is the most precise and stringently tested theory in physics. The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac , who (during the 1920s) was able to compute the coefficient of spontaneous emission of an atom . He

4464-411: The electron orbital g-factor , g L , is defined by μ L = − g L μ B ℏ L , {\displaystyle {\boldsymbol {\mu }}_{L}=-g_{L}{\mu _{\mathrm {B} } \over \hbar }\mathbf {L} ,} where μ L is the magnetic moment resulting from the orbital angular momentum of an electron, L

4557-446: The probability of the event is the sum of the probabilities of the alternatives. Indeed, if this were not the case, the very term "alternatives" to describe these processes would be inappropriate. What (a) says is that once the physical means for observing which alternative occurred is removed , one cannot still say that the event is occurring through "exactly one of the alternatives" in the sense of adding probabilities; one must add

4650-576: The quantum Hall effect . From these two constants, the elementary charge can be deduced: e = 2 R K K J . {\displaystyle e={\frac {2}{R_{\text{K}}K_{\text{J}}}}.} The relation used by CODATA to determine elementary charge was: e 2 = 2 h α μ 0 c = 2 h α ε 0 c , {\displaystyle e^{2}={\frac {2h\alpha }{\mu _{0}c}}=2h\alpha \varepsilon _{0}c,} where h

4743-529: The weak nuclear force and quantum electrodynamics could be merged into a single electroweak force . Near the end of his life, Richard Feynman gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), QED: The Strange Theory of Light and Matter , a classic non-mathematical exposition of QED from the point of view articulated below. The key components of Feynman's presentation of QED are three basic actions. These actions are represented in

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4836-416: The " quantum of charge" is ⁠ 1 / 3 ⁠   e . In this case, one says that the "elementary charge" is three times as large as the "quantum of charge". On the other hand, all isolatable particles have charges that are integer multiples of e . (Quarks cannot be isolated: they exist only in collective states like protons that have total charges that are integer multiples of e .) Therefore,

4929-403: The "quantum of charge" is e , with the proviso that quarks are not to be included. In this case, "elementary charge" would be synonymous with the "quantum of charge". In fact, both terminologies are used. For this reason, phrases like "the quantum of charge" or "the indivisible unit of charge" can be ambiguous unless further specification is given. On the other hand, the term "elementary charge"

5022-534: The Avogadro constant N A was first approximated by Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas. Today the value of N A can be measured at very high accuracy by taking an extremely pure crystal (often silicon ), measuring how far apart the atoms are spaced using X-ray diffraction or another method, and accurately measuring

5115-540: The Lagrangian contains no ∂ μ ψ ¯ {\displaystyle \partial _{\mu }{\bar {\psi }}} terms, we immediately get so the equation of motion can be written ( i γ μ ∂ μ − m ) ψ = e γ μ A μ ψ . {\displaystyle (i\gamma ^{\mu }\partial _{\mu }-m)\psi =e\gamma ^{\mu }A_{\mu }\psi .}

5208-414: The air), and electric force . The forces due to gravity and viscosity could be calculated based on the size and velocity of the oil drop, so electric force could be deduced. Since electric force, in turn, is the product of the electric charge and the known electric field, the electric charge of the oil drop could be accurately computed. By measuring the charges of many different oil drops, it can be seen that

5301-438: The alternatives for E and F . (This is not elementary in practice and involves integration .) But there is another possibility, which is that the electron first moves to G , where it emits a photon, which goes on to D , while the electron moves on to H , where it absorbs the first photon, before moving on to C . Again, we can calculate the probability amplitude of these possibilities (for all points G and H ). We then have

5394-456: The amplitudes instead. Similarly, the independence criterion in (b) is very important: it only applies to processes which are not "entangled". Suppose we start with one electron at a certain place and time (this place and time being given the arbitrary label A ) and a photon at another place and time (given the label B ). A typical question from a physical standpoint is: "What is the probability of finding an electron at C (another place and

5487-559: The associated quantity is written in Feynman's shorthand as P ( A  to  B ) {\displaystyle P(A{\text{ to }}B)} , and it depends on only the momentum and polarization of the photon. The similar quantity for an electron moving from C {\displaystyle C} to D {\displaystyle D} is written E ( C  to  D ) {\displaystyle E(C{\text{ to }}D)} . It depends on

5580-475: The basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above ( P ( A to B ), E ( C to D ) and j ) acts just like our everyday probability (a simplification made in Feynman's book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman. The basic rules of probability amplitudes that will be used are: The indistinguishability criterion in (a)

5673-428: The charges are all integer multiples of a single small charge, namely e . The necessity of measuring the size of the oil droplets can be eliminated by using tiny plastic spheres of a uniform size. The force due to viscosity can be eliminated by adjusting the strength of the electric field so that the sphere hovers motionless. Any electric current will be associated with noise from a variety of sources, one of which

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5766-408: The covariant derivative reveals a second useful form of the Lagrangian (external field B μ {\displaystyle B_{\mu }} set to zero for simplicity) where j μ {\displaystyle j^{\mu }} is the conserved U ( 1 ) {\displaystyle {\text{U}}(1)} current arising from Noether's theorem. It

5859-478: The density of the crystal. From this information, one can deduce the mass ( m ) of a single atom; and since the molar mass ( M ) is known, the number of atoms in a mole can be calculated: N A = M / m . The value of F can be measured directly using Faraday's laws of electrolysis . Faraday's laws of electrolysis are quantitative relationships based on the electrochemical researches published by Michael Faraday in 1834. In an electrolysis experiment, there

5952-1014: The derivatives this time are ∂ ν ( ∂ L ∂ ( ∂ ν A μ ) ) = ∂ ν ( ∂ μ A ν − ∂ ν A μ ) , {\displaystyle \partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }A_{\mu })}}\right)=\partial _{\nu }\left(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }\right),} ∂ L ∂ A μ = − e ψ ¯ γ μ ψ . {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{\mu }}}=-e{\bar {\psi }}\gamma ^{\mu }\psi .} Substituting back into ( 3 ) leads to which can be written in terms of

6045-404: The electron can be polarized is another small necessary detail, which is connected with the fact that an electron is a fermion and obeys Fermi–Dirac statistics . The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include (as we always must) the complementary Feynman diagram in which we exchange two electron events,

6138-399: The electron, has a g -factor associated with its spin, given by the equation μ = g e 2 m μ S , {\displaystyle {\boldsymbol {\mu }}=g{e \over 2m_{\mu }}\mathbf {S} ,} where μ is the magnetic moment resulting from the muon's spin, S is the spin angular momentum, and m μ is the muon mass. That

6231-447: The elementary charge in 1909, differing from the modern accepted value by just 0.6%. Under assumptions of the then-disputed atomic theory , the elementary charge had also been indirectly inferred to ~3% accuracy from blackbody spectra by Max Planck in 1901 and (through the Faraday constant ) at order-of-magnitude accuracy by Johann Loschmidt 's measurement of the Avogadro number in 1865. In some natural unit systems, such as

6324-675: The equations reduce to ◻ A μ = e j μ , {\displaystyle \Box A^{\mu }=ej^{\mu },} which is a wave equation for the four-potential, the QED version of the classical Maxwell equations in the Lorenz gauge . (The square represents the wave operator , ◻ = ∂ μ ∂ μ {\displaystyle \Box =\partial _{\mu }\partial ^{\mu }} .) This theory can be straightforwardly quantized by treating bosonic and fermionic sectors as free. This permits us to build

6417-481: The experimental measured value is 2.002 331 8416 (13) , compared to the theoretical prediction of 2.002 331 836 20 (86) . This is a difference of 3.4 standard deviations , suggesting that beyond-the-Standard-Model physics may be a contributory factor. The Brookhaven muon storage ring was transported to Fermilab where the Muon g –2 experiment used it to make more precise measurements of muon g -factor. On April 7, 2021,

6510-496: The field-theoretic, operator -based approach of Schwinger and Tomonaga, but Freeman Dyson later showed that the two approaches were equivalent. Renormalization , the need to attach a physical meaning at certain divergences appearing in the theory through integrals , has subsequently become one of the fundamental aspects of quantum field theory and has come to be seen as a criterion for a theory's general acceptability. Even though renormalization works very well in practice, Feynman

6603-408: The form of visual shorthand by the three basic elements of diagrams : a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram. As well as the visual shorthand for the actions, Feynman introduces another kind of shorthand for

6696-474: The formula e = F N A . {\displaystyle e={\frac {F}{N_{\text{A}}}}.} (In other words, the charge of one mole of electrons, divided by the number of electrons in a mole, equals the charge of a single electron.) This method is not how the most accurate values are measured today. Nevertheless, it is a legitimate and still quite accurate method, and experimental methodologies are described below. The value of

6789-400: The momentum and polarization of the electron, in addition to a constant Feynman calls n , sometimes called the "bare" mass of the electron: it is related to, but not the same as, the measured electron mass. Finally, the quantity that tells us about the probability amplitude for an electron to emit or absorb a photon Feynman calls j , and is sometimes called the "bare" charge of the electron: it

6882-488: The muon g -factor is not quite the same as the electron g -factor is mostly explained by quantum electrodynamics and its calculation of the anomalous magnetic dipole moment. Almost all of the small difference between the two values (99.96% of it) is due to a well-understood lack of heavy-particle diagrams contributing to the probability for emission of a photon representing the magnetic dipole field, which are present for muons, but not electrons, in QED theory. These are entirely

6975-552: The new field as A μ . {\displaystyle A_{\mu }.} From this Lagrangian, the equations of motion for the ψ {\displaystyle \psi } and A μ {\displaystyle A_{\mu }} fields can be obtained. These arise most straightforwardly by considering the Euler-Lagrange equation for ψ ¯ {\displaystyle {\bar {\psi }}} . Since

7068-478: The numerical quantities called probability amplitudes . The probability is the square of the absolute value of total probability amplitude, probability = | f ( amplitude ) | 2 {\displaystyle {\text{probability}}=|f({\text{amplitude}})|^{2}} . If a photon moves from one place and time A {\displaystyle A} to another place and time B {\displaystyle B} ,

7161-414: The points to which a particle can move, that is not true in full quantum electrodynamics. There is a nonzero probability amplitude of an electron at A , or a photon at B , moving as a basic action to any other place and time in the universe . That includes places that could only be reached at speeds greater than that of light and also earlier times . (An electron moving backwards in time can be viewed as

7254-436: The precision of the method is the measurement of F : the best experimental value has a relative uncertainty of 1.6 ppm, about thirty times higher than other modern methods of measuring or calculating the elementary charge. A famous method for measuring e is Millikan's oil-drop experiment. A small drop of oil in an electric field would move at a rate that balanced the forces of gravity , viscosity (of traveling through

7347-437: The probability amplitude for an electron to get from A to B , we must take into account all the possible ways: all possible Feynman diagrams with those endpoints. Thus there will be a way in which the electron travels to C , emits a photon there and then absorbs it again at D before moving on to B . Or it could do this kind of thing twice, or more. In short, we have a fractal -like situation in which if we look closely at

7440-507: The probability amplitudes for the photon and the electron respectively. These are essentially the solutions of the Dirac equation , which describe the behavior of the electron's probability amplitude and the Maxwell's equations , which describes the behavior of the photon's probability amplitude. These are called Feynman propagators . The translation to a notation commonly used in the standard literature

7533-405: The probability amplitudes of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption. We would expect to find the total probability amplitude by multiplying the probability amplitudes of each of the actions, for any chosen positions of E and F . We then, using rule a) above, have to add up all these probability amplitudes for all

7626-421: The proton's mass rather than the particle's mass as for a Dirac particle. The formula used under this convention is μ = g μ N ℏ I = g e 2 m p I , {\displaystyle {\boldsymbol {\mu }}=g{\mu _{\text{N}} \over \hbar }{\mathbf {I} }=g{e \over 2m_{\text{p}}}\mathbf {I} ,} where μ

7719-402: The result that e = 4 π α ε 0 ℏ c ≈ 0.30282212088 ε 0 ℏ c , {\displaystyle e={\sqrt {4\pi \alpha }}{\sqrt {\varepsilon _{0}\hbar c}}\approx 0.30282212088{\sqrt {\varepsilon _{0}\hbar c}},} where α is the fine-structure constant , c

7812-458: The resulting amplitude is the reverse – the negative – of the first. The simplest case would be two electrons starting at A and B ending at C and D . The amplitude would be calculated as the "difference", E ( A to D ) × E ( B to C ) − E ( A to C ) × E ( B to D ) , where we would expect, from our everyday idea of probabilities, that it would be a sum. Finally, one has to compute P ( A to B ) and E ( C to D ) corresponding to

7905-534: The simple rule that the probability of an event is the square of the length of the corresponding amplitude arrow. So, for a given process, if two probability amplitudes, v and w , are involved, the probability of the process will be given either by or The rules as regards adding or multiplying, however, are the same as above. But where you would expect to add or multiply probabilities, instead you add or multiply probability amplitudes that now are complex numbers. Addition and multiplication are common operations in

7998-416: The system of atomic units , e functions as the unit of electric charge . The use of elementary charge as a unit was promoted by George Johnstone Stoney in 1874 for the first system of natural units, called Stoney units . Later, he proposed the name electron for this unit. At the time, the particle we now call the electron was not yet discovered and the difference between the particle electron and

8091-435: The theory increased through the end of the 1940s. Improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom , now known as the Lamb shift and magnetic moment of the electron. These experiments exposed discrepancies which the theory was unable to explain. A first indication of a possible way out was given by Hans Bethe in 1947, after attending

8184-415: The theory of complex numbers and are given in the figures. The sum is found as follows. Let the start of the second arrow be at the end of the first. The sum is then a third arrow that goes directly from the beginning of the first to the end of the second. The product of two arrows is an arrow whose length is the product of the two lengths. The direction of the product is found by adding the angles that each of

8277-423: The two have been turned through relative to a reference direction: that gives the angle that the product is turned relative to the reference direction. That change, from probabilities to probability amplitudes, complicates the mathematics without changing the basic approach. But that change is still not quite enough because it fails to take into account the fact that both photons and electrons can be polarized, which

8370-476: The unit of charge electron was still blurred. Later, the name electron was assigned to the particle and the unit of charge e lost its name. However, the unit of energy electronvolt (eV) is a remnant of the fact that the elementary charge was once called electron . In other natural unit systems, the unit of charge is defined as ε 0 ℏ c , {\displaystyle {\sqrt {\varepsilon _{0}\hbar c}},} with

8463-402: Was "fixed" by the technique of renormalization . However, Feynman himself remained unhappy about it, calling it a "dippy process", and Dirac also criticized this procedure as "in mathematics one does not get rid of infinities when it does not please you". Within the above framework physicists were then able to calculate to a high degree of accuracy some of the properties of electrons, such as

8556-400: Was never entirely comfortable with its mathematical validity, even referring to renormalization as a "shell game" and "hocus pocus". Thence, neither Feynman nor Dirac were happy with that way to approach the observations made in theoretical physics, above all in quantum mechanics. QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory

8649-527: Was used in the first direct observation of Laughlin quasiparticles , implicated in the fractional quantum Hall effect . Another accurate method for measuring the elementary charge is by inferring it from measurements of two effects in quantum mechanics : The Josephson effect , voltage oscillations that arise in certain superconducting structures; and the quantum Hall effect , a quantum effect of electrons at low temperatures, strong magnetic fields, and confinement into two dimensions. The Josephson constant

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