Σχετικιστικές Κυματικές Εξισώσεις

Σχετικιστικοί Κυματικοί Νόμοι

Relativistic wave equations, Laws of physics

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Διαφορική Εξίσωση
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- Νόμοι της Φυσικής.

- Ακριβέστερα, είναι νόμοι της Σχετικιστικής Φυσικής και της Κυματικής

- Χρονολογία ανακάλυψης.

Ετυμολογία

Η ονομασία "σχετικιστική" σχετίζεται ετυμολογικά με την λέξη "Σχετικότητα".

Διατύπωση

In physics, specifically relativistic quantum mechanics (RQM) and its applications to particle physics, relativistic wave equations predict the behavior of particles at high energies and velocities comparable to the speed of light. In the context of quantum field theory (QFT), the equations determine the dynamics of quantum fields.

The solutions to the equations, universally denoted as ψ or Ψ (Greek psi), are referred to as "wavefunctions" in the context of RQM, and "fields" in the context of QFT. The equations themselves are called "wave equations" or "field equations", because they have the mathematical form of a wave equation or are generated from a Lagrangian (density) and the Euler–Lagrange equations for a field (see classical field theory for background).

In each case; the solution, the wavefunction or field, can be inferred from the attributes of a quantum state vector (an element of a Hilbert space), universally denoted in bra–ket notation by |ψΠρότυπο:Rangle or |ΨΠρότυπο:Rangle. (See wavefunction and bra–ket notation for how this is done). In the Schrödinger picture, the time evolution of this vector describing the system, in turn the dynamics of the wave or field, is given by the Schrödinger equation;

$i\hbar {\frac {d}{dt}}\left|\psi \right\rangle ={\hat {H}}\left|\psi \right\rangle \quad \rightleftharpoons \quad i\hbar {\frac {\partial }{\partial t}}\psi ={\hat {H}}\psi$

(the first is an abstract linear algebra form, the second is in terms of calculus), one of the postulates of quantum mechanics, by specifying various forms of the Hamiltonian operator Ĥ. All relativistic wave equations can be constructed by choosing different forms of the Hamiltonian. Alternatively, Feynman's path integral formulation uses a Lagrangian rather than a Hamiltonian operator.

More generally - the modern formalism behind relativistic wave equations is Lorentz group theory, wherein the spin of the particle has a correspondence with the representations of the Lorentz group.

History

Early 1920s: Classical and quantum mechanics

The failure of classical mechanics (applied to molecular, atomic, and nuclear systems and smaller) induced the need for a new mechanics: quantum mechanics. The mathematical formulation was led by De Broglie, Bohr, Schrödinger, Pauli, and Heisenberg, and others, around the mid-1920s, and at that time was analogous to that of classical mechanics. The Schrödinger equation and the Heisenberg picture resemble the classical equations of motion in the limit of large quantum numbers and as the reduced Planck constant ħ, the quantum of action, tends to zero. This is the correspondence principle. For this reason, the Schrödinger and Heisenberg formulations are non-relativistic, so they can't be used in situations where the particles travel near the speed of light, or when the number of each type of particle changes (which happens in real particle interactions; the numerous forms of particle decays, annihilation, matter creation, pair production, and so on).

Late 1920s: Relativistic quantum mechanics of spin-0 and spin-½ particles

A relativistic description of quantum mechanical systems was sought for by many theoretical physicists; from the late 1920s to the mid-1940s. The first quantum-relativistic basis was found by all those who discovered what is frequently called the Klein–Gordon equation:

Πρότυπο:NumBlk

by inserting the energy operator and momentum operator into the relativistic energy–momentum relation:

Πρότυπο:NumBlk

The solutions to (Πρότυπο:EquationNote) are scalar fields. The KG equation is undesirable due to its prediction of negative energies and probabilities, as a result of the quadratic nature of (Πρότυπο:EquationNote) - inevitable in a relativistic theory. This equation was proposed initially proposed by Schrödinger, and he discarded it for such reasons, only to realize a few months later that its non-relativistic limit (what is now called the Schrödinger equation) was still of importance. Nevertheless - (Πρότυπο:EquationNote) is applicable to spin-0 bosons.

Neither the non-relativistic nor relativistic equations found by Schrödinger could predict the hyperfine structure in the Hydrogen spectral series. The mysterious underlying property was spin. The first two-dimensional spin matrices (better known as the Pauli matrices) were introduced by Pauli in the Pauli equation; the Schrödinger equation with a non-relativistic Hamiltonian including an extra term for particles in magnetic fields, but this was phenomological. Weyl found a relativistic equation in terms of the Pauli matrices; the Weyl equation, for massless spin-½ fermions. The problem was resolved by Dirac in the late 1920s, when he furthered the application of equation (Πρότυπο:EquationNote) to the electron - by various manipulations he factorized the equation into the form:

Πρότυπο:NumBlk

and one of these factors is the Dirac equation (see below), upon inserting the energy and momentum operators. For the first time, this introduced new four-dimensional spin matrices α and β in a relativistic wave equation, and explained the hyperfine structure of hydrogen. The solutions to (Πρότυπο:EquationNote) are multi-component spinor fields, and each component satisfies (Πρότυπο:EquationNote). A remarkable result of spinor solutions is that half of the components describe a particle, while the other half describe an antiparticle; in this case the electron and positron. The Dirac equation is now known to apply for all massive spin-½ fermions. In the non-relativistic limit, the Pauli equation is recovered, while the massless case results in the Weyl equation.

Although a landmark in quantum theory, the Dirac equation is only true for spin-½ fermions, and still predicts negative energy solutions, which caused controversy at the time (in particular - not all physicists were comfortable with the "Dirac sea" of negative energy states).

1930s–1960s: Relativistic quantum mechanics of higher-spin particles

The natural problem became clear: to generalize the Dirac equation to particles with any spin; both fermions and bosons, and in the same equations their antiparticles (possible because of the spinor formalism introduced by Dirac in his equation, and then-recent developments in spinor calculus by van der Waerden in 1929), and ideally with positive energy solutions.

This was introduced and solved by Majorana in 1932, by a deviated approach to Dirac. Majorana considered one "root" of (Πρότυπο:EquationNote):

Πρότυπο:NumBlk

where ψ is a spinor field now with infinitely many components, irreducible to a finite number of tensors or spinors, to remove the indeterminacy in sign. The matrices α and β are infinite-dimensional matrices, related to infinitesimal Lorentz transformations. He did not demand that each component of to satisfy equation (Πρότυπο:EquationNote), instead he regenerated the equation using a Lorentz-invariant action, via the principle of least action, and application of Lorentz group theory.

Majorana produced other important contributions that were unpublished, including wave equations of various dimensions (5, 6, and 16). They were anticipated later (in a more involved way) by de Broglie (1934), and Duffin, Kemmer, and Petiau (around 1938–1939), see Duffin–Kemmer–Petiau algebra. The Dirac–Fierz–Pauli formalism was more sophisticated than Majorana’s, as spinors were new mathematical tools in the early twentieth century, although Majorana’s paper of 1932 was difficult to fully understand; it took Pauli and Wigner took some time to understand it, around 1940.

Dirac in 1936, and Fierz and Pauli in 1939, built equations from irreducible spinors A and B, symmetric in all indices, for a massive particle of spin n + ½ for integer n (see Van der Waerden notation for the meaning of the dotted indices):

Πρότυπο:NumBlk

Πρότυπο:NumBlk

where p is the momentum as a covariant spinor operator. For n = 0, the equations reduce to the coupled Dirac equations and A and B together transform as the original Dirac spinor. Eliminating either A or B shows that A and B each fulfill (Πρότυπο:EquationNote).

In 1941, Rarita and Schwinger focussed on spin-Πρότυπο:Frac particles and derived the Rarita–Schwinger equation, including a Lagrangian to generate it, and later generalized the equations analogous to spin n + ½ for integer n. In 1945, Pauli suggested Majorana's 1932 paper to Bhabha, who returned to the general ideas introduced by Majorana in 1932. Bhabha and Lubanski proposed a completely general set of equations by replacing the mass terms in (Πρότυπο:EquationNote) and (Πρότυπο:EquationNote) by an arbitrary constant, subject to a set of conditions which the wavefunctions must obey.

Finally, in the year 1948 (the same year as Feynman's path integral formulation was cast), Bargmann and Wigner formulated the general equation for massive particles which could have any spin, by considering the Dirac equation with a totally symmetric finite-component spinor, and using Lorentz group theory (as Majorana did): the Bargmann–Wigner equations. In the early 1960s, a reformulation of the Bargmann–Wigner equations was made by H. Joos and Steven Weinberg. Various theorists at this time did further research in relativistic Hamiltonians for higher spin particles.

1960s–Present

The relativistic description of spin particles has been a difficult problem in quantum theory. It is still an area of present-day research, because the problem is only partially solved; including interactions in the equations is problematic, and paradoxical predictions (even from the Dirac equation) are still present.

Linear equations

The following equations have solutions which satisfy the superposition principle, that is, the wavefunctions are additive.

Throughout, tensor index notation and Feynman slash notation is used. In matrix equations, the gamma matrices are denoted by γ^μ, and the Pauli matrices by σ^μ (where σ⁰ is the 2 × 2 identity matrix). The expression (iħγ^μ∂_μ + mc) is a 4 × 4 matrix operator which acts on the spinor fields (denoted ψ), it contains the four-gradient operator ∂_μ. Terms such as "mc" scalar multiply an identity matrix of the relevant dimension, the common sizes are 2 × 2 or 4 × 4, and are conventionally not written for simplicity.

Particle spin quantum number s	Name	Equation	Typical particles the equation describes
0	Εξίσωση Klein-Gordon	$(\hbar \partial _{\mu }+imc)(\hbar \partial ^{\mu }-imc)\psi =0$	Massless or massive spin-0 particle (such as Higgs bosons).
1/2	Weyl equation	$\sigma ^{\mu }\partial _{\mu }\psi =0$	Massless spin-1/2 particles.
	Dirac equation	$\left(i\hbar \partial \!\!\!/-mc\right)\psi =0$	Massive spin-1/2 particles (such as electrons).
	Two-body Dirac equations	$[(\gamma _{1})_{\mu }(p_{1}-{\tilde {A}}_{1})^{\mu }+m_{1}+{\tilde {S}}_{1}]\Psi =0,$ $[(\gamma _{2})_{\mu }(p_{2}-{\tilde {A}}_{2})^{\mu }+m_{2}+{\tilde {S}}_{2}]\Psi =0.$	Massive spin-1/2 particles (such as electrons).
	Εξίσωση Majorana	$i\hbar \partial \!\!\!/\psi -mc\psi _{c}=0$	Massive Majorana particles.
	Breit equation	$i\hbar {\frac {\partial \Psi }{\partial t}}=\left(\sum _{i}{\hat {H}}_{D}(i)+\sum _{i>j}{\frac {1}{r_{ij}}}-\sum _{i>j}{\hat {B}}_{ij}\right)\Psi$	Two massive spin-1/2 particles (such as electrons) interacting electromagnetically to first order in perturbation theory.
1	Εξισώσεις Maxwell (in QED using the Lorenz gauge)	$\partial _{\mu }\partial ^{\mu }A^{\nu }=e{\overline {\psi }}\gamma ^{\nu }\psi$	Photons, massless spin-1 particles.
	Proca equation	$\partial _{\mu }(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })+\left({\frac {mc}{\hbar }}\right)^{2}A^{\nu }=0$	Massive spin-1 particle (such as W and Z bosons).
3/2	Εξίσωση Rarita-Schwinger	$\epsilon ^{\mu \nu \rho \sigma }\gamma ^{5}\gamma _{\nu }\partial _{\rho }\psi _{\sigma }+m\psi ^{\mu }=0$	Massive spin-3/2 particles.
s	Εξισώσεις Bargmann-Wigner	${\begin{aligned}&(-i\hbar \gamma ^{\mu }\partial _{\mu }+mc)_{\alpha _{1}\alpha _{1}'}\psi _{\alpha '_{1}\alpha _{2}\alpha _{3}\cdots \alpha _{2s}}=0\\&(-i\hbar \gamma ^{\mu }\partial _{\mu }+mc)_{\alpha _{2}\alpha _{2}'}\psi _{\alpha _{1}\alpha '_{2}\alpha _{3}\cdots \alpha _{2s}}=0\\&\qquad \vdots \\&(-i\hbar \gamma ^{\mu }\partial _{\mu }+mc)_{\alpha _{2s}\alpha '_{2s}}\psi _{\alpha _{1}\alpha _{2}\alpha _{3}\cdots \alpha '_{2s}}=0\\\end{aligned}}$ where ψ is a rank-2s 4-component spinor.	Free particles of arbitrary spin (bosons and fermions).

Gauge fields

The Duffin-Kemmer-Petiau equation is an alternative equation for spin-0 and spin-1 particles:

$(i\hbar \beta ^{a}\partial _{a}-mc)\psi =0$

Non-linear equations

Πρότυπο:Further

There are equations which have solutions that do not satisfy the superposition principle.

Gauge fields

• Yang-Mills equation: describes a non-abelian gauge field
• Yang-Mills-Higgs equations: describes a non-abelian gauge field coupled with a massive spin-0 particle

Spin 2

• Einstein field equations: describe interaction of matter with the gravitational field (massless spin-2 field):

$R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}$

The solution is a metric tensor field, rather than a wavefunction.

Εσωτερική Αρθρογραφία

• Σχετικιστική Κυματική Εξίσωση
• Scalar field theory
• special relativity
• electromagnetic field
• Minimal coupling
• Lorentz transformations
• quantum mechanics
• nuclear Physics, particle physics
• Επιστημονικός Νόμος
• Επιστημονικοί Νόμοι
• Εξίσωση
• Εξισώσεις
• Φυσικός Νόμος
• Φυσικοί Νόμοι
• Νόμοι Διατήρησης
• Νόμοι Νεύτωνα
• Εξισώσεις Maxwell
• Ηλεκτρικός Νόμος Gauss
• Μαγνητικός Νόμος Gauss
• Νόμος Faraday
• Θεώρημα Απόκλισης
• Μαγνητική Ροή

Ιστογραφία

• Ομώνυμο άρθρο στην Βικιπαίδεια
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Κίνδυνοι Χρήσης

Αν και θα βρείτε εξακριβωμένες πληροφορίες σε αυτήν την εγκυκλοπαίδεια ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη, την εγκυρότητα των πληροφοριών που περιλαμβάνει. "Οι πληροφορίες αυτές μπορεί πρόσφατα να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο, η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης" του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει." Πρέπει να λάβετε υπ' όψη ότι όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς, και για μακρά χρονική περίοδο, αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία, ελάχιστο χρονικό διάστημα, πριν τα δείτε. Επίσης, Οι διάφοροι "Εξωτερικοί Σύνδεσμοι (Links)" (όχι μόνον, της Sciencepedia αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)), αν και άκρως απαραίτητοι, είναι αδύνατον να ελεγχθούν (λόγω της ρευστής φύσης του Web), και επομένως είναι ενδεχόμενο να οδηγήσουν σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο. Ο αναγνώστης πρέπει να είναι εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

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