Four-gradient: Difference between revisions
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=== As a component of the covariant form of the quantum commutation relation === |
=== As a component of the covariant form of the quantum commutation relation === |
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In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). |
In quantum mechanics (physics), the [[canonical commutation relation]] is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). |
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<math>[P^\mu,X^\nu] = i \hbar [\partial^\mu,X^\nu] = i \hbar \eta^{\mu \nu}</math><ref>{{cite book |
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=== As a component of the wave equations and probability currents in relativistic quantum mechanics === |
=== As a component of the wave equations and probability currents in relativistic quantum mechanics === |
Revision as of 12:59, 31 May 2016
In differential geometry, the four-gradient (4-gradient) is the four-vector (4-vector) analogue of the gradient from Gibbs-Heaviside vector calculus.
In special relativity and in quantum mechanics, the 4-gradient is used to define the properties and relations between the various physical 4-vectors and tensors.
This article uses the (+---) metric signature and tensor index notation in the language of 4-vectors.
Definition
The covariant components compactly written in tensor index notation are:[1] [2]
The comma in the last part above implies the partial differentiation with respect to . This is not the same as a semi-colon, used for the covariant derivative.
The contravariant components are:[3] [4]
where gaß is the metric tensor, which here has been chosen for flat spacetime with the metric signature (+,-,-,-).
Alternative symbols to are and D.
Usage
The 4-gradient is used in a number of different ways in special relativity (SR):
Throughout this article the formulas are correct for Minkowski coordinates in SR, but may need to be modified for other coordinates.
There are alternate ways of writing the expressions:
- is a 4-vector style, which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the 4-vector.
- is a tensor index style, which is sometimes required in more complicated expressions, especially those involving tensors with more than one index, such as .
- The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):
As a 4-divergence
Divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point.
The 4-divergence of the 4-position gives the dimension of spacetime:
The 4-divergence of the 4-current gives a conservation law - the conservation of charge:[5]
This means that the time rate of change of the charge density must equal the negative spatial divergence of the current density .
In other words, the charge inside a box cannot just change arbitrarily, it must enter and leave the box via a current. This is a continuity equation.
The 4-divergence of the electromagnetic 4-potential is used in the Lorenz gauge condition:[6]
This is the equivalent of a conservation law for the EM 4-potential.
As a Jacobian matrix for the SR metric tensor
The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
The 4-gradient acting on the 4-position gives the SR Minkowski space metric :[7]
For the Minkowski metric, the components
As part of the total proper time derivative
The scalar product of 4-velocity with the 4-gradient gives the total derivative with respect to proper time :[8]
The fact that is a Lorentz scalar invariant shows that the total derivative with respect to proper time is likewise a Lorentz scalar invariant.
So, for example, the 4-velocity is the proper-time derivative of the 4-position :
or
Another example, the 4-acceleration is the proper-time derivative of the 4-velocity :
or
As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations
The Faraday electromagnetic tensor is a mathematical object that describes the electromagnetic field in space-time of a physical system.[9] [10] [11] [12] [13]
Applying the 4-gradient to make an antisymmetric tensor, one gets:
where:
- Electromagnetic 4-potential , not to be confused with the 4-acceleration
is the electric scalar potential, and is the magnetic 3-vector potential.
By applying the 4-gradient again, and defining the 4-current as one can derive the tensor form of the Maxwell equations:
where the second line is a version of the Bianchi identity.
As a way to define the 4-wavevector
A wavevector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave (inversely proportional to the wavelength), and its direction is ordinarily the direction of wave propagation
The 4-wavevector is the 4-gradient of the negative phase (or the negative 4-gradient of the phase) of a wave in Minkowski Space: [14]
This is mathematically equivalent to the definition of the phase of a wave (or more specifically a plane wave):
where 4-position , is the temporal angular frequency, is the spatial 3-wavevector, and is the Lorentz scalar invariant phase.
with the assumption that the plane wave and are not explicit functions of or
The explicit form of the plane wave can be written as:
- where is a (possibly complex) amplitude.
A general wave would be the superposition of multiple plane waves:
As the d'Alembertian operator
The square of is the 4-Laplacian, which is called the d'Alembert operator:[15] [16] [17]
- .
As it is the dot product of two 4-vectors, the d'Alembertian is a Lorentz invariant scalar.
Occasionally, in analogy with the 3-dimensional notation, the symbols and are used for the 4-gradient and d'Alembertian respectively. More commonly however, the symbol is reserved for the d'Alembertian.
Some examples of the 4-gradient as used in the d'Alembertian follow:
In the Klein-Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs_boson):
In the wave equation for the electromagnetic field { using Lorenz gauge }:
- {in vacuum}
- {with a 4-current source}
where:
- Electromagnetic 4-potential is an electromagnetic vector potential
- 4-current is an electromagnetic current density
In the 4-dimensional version of Green's function:
As a component of the Schrödinger relations in quantum mechanics
The 4-gradient is connected with quantum mechanics.
The relation between the 4-momentum and the 4-gradient gives the Schrödinger QM relations.[18]
The temporal component gives:
The spatial components give:
This can actually be composed of two separate steps.
First:[19]
which is the full 4-vector version of:
The (temporal component) Planck–Einstein relation
The (spatial components) de Broglie matter wave relation
Second:[20]
which is just the 4-gradient version of the wave equation for complex-valued plane waves
The temporal component gives:
The spatial components give:
As a component of the covariant form of the quantum commutation relation
In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).
As a component of the wave equations and probability currents in relativistic quantum mechanics
The 4-gradient is a component in several of the relativistic wave equations:[22] [23]
In the Klein-Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs_boson):
In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons):
where are the Dirac gamma matrices and is a relativistic wave function.
is Lorentz scalar for the Klein-Gordon equation, and a spinor for the Dirac equation.
It is nice that the gamma matrices themselves refer back to the fundamental aspect of SR, the Minkowski metric:
Conservation of 4-probability current density follows from the continuity equation:
The 4-probability current density has the relativistically covariant expression:
As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity
Start with the standard SR 4-vectors:
- 4-position
- 4-velocity
- 4-momentum
- 4-wavevector
- 4-gradient
Note the following simple relations from the previous sections, where each 4-vector is related to another by a Lorentz scalar:
- , where is the proper time
- , where is the rest mass
- , which is the 4-vector version of the Planck-Einstein relation & the de Broglie matter wave relation
- , which is the gradient version of complex-valued plane waves
Now, just apply the standard Lorentz scalar product rule to each one:
The last equation (with the 4-gradient scalar product) is a fundamental quantum relation.
When applied to a Lorentz scalar field , one gets the Klein-Gordon equation, the most basic of the quantum relativistic wave equations.
The Schrödinger equation is the low-velocity limiting case (v<<c) of the Klein-Gordon equation.
If the last part is applied to a 4-vector field instead of a Lorentz scalar field , then one gets the Proca equation:
If the rest mass term is set to zero (light-like particles), then this gives the free Maxwell equation:
More complicated forms and interactions can be derived by using the minimal coupling rule:
Derivation
In three dimensions, the gradient operator maps a scalar field to a vector field such that the line integral between any two points in the vector field is equal to the difference between the scalar field at these two points. Based on this, it may appear incorrectly that the natural extension of the gradient to 4 dimensions should be:
incorrect
However, a line integral involves the application of the vector dot product, and when this is extended to 4-dimensional space-time, a change of sign is introduced to either the spatial co-ordinates or the time co-ordinate depending on the convention used. This is due to the non-Euclidean nature of space-time. In this article, we place a negative sign on the spatial coordinates (the time-positive Metric convention ). The factor of (1/c) is to keep the correct unit dimensionality {1/[length]} for all components of the 4-vector and the (-1) is to keep the 4-gradient Lorentz covariant. Adding these two corrections to the above expression gives the correct definition of 4-gradient:
correct
See also
- four-vector
- Ricci calculus
- Index notation
- Tensor
- Antisymmetric tensor
- Einstein notation
- Raising and lowering indices
- Abstract index notation
- Covariance and contravariance of vectors
References
- ^ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 16. ISBN 0-201-62460-5.
- ^ The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 16. ISBN 0-201-62460-5.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. p. 103-107. ISBN 0-19-853952-5.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. p. 105-107. ISBN 0-19-853952-5.
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 16. ISBN 0-201-62460-5.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. p. 58-59. ISBN 0-19-853952-5.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. p. 101-128. ISBN 0-19-853952-5.
- ^ Sudbury, Anthony (1986). Quantum mechanics and the particles of nature: An outline for mathematicians (1st ed.). Cambridge University Press. p. 314. ISBN 0-521-27765-5.
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 17-18. ISBN 0-201-62460-5.
- ^ Carroll, Sean M. (2004). An Introduction to General Relativity: Spacetime and Geometry (1st ed.). Addison-Wesley Publishing Co. p. 29-30. ISBN 0-8053-8732-3.
- ^ Greiner, Walter (2000). Relativistic Quantum Mechanics: Wave Equations (3rd ed.). Springer. p. 4. ISBN 3-540-67457-8.
- ^ Carroll, Sean M. (2004). An Introduction to General Relativity: Spacetime and Geometry (1st ed.). Addison-Wesley Publishing Co. p. 387. ISBN 0-8053-8732-3.
- ^ Sudbury, Anthony (1986). Quantum mechanics and the particles of nature: An outline for mathematicians (1st ed.). Cambridge University Press. p. 300. ISBN 0-521-27765-5.
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 17-18. ISBN 0-201-62460-5.
- ^ Carroll, Sean M. (2004). An Introduction to General Relativity: Spacetime and Geometry (1st ed.). Addison-Wesley Publishing Co. p. 41. ISBN 0-8053-8732-3.
- ^ Greiner, Walter (2000). Relativistic Quantum Mechanics: Wave Equations (3rd ed.). Springer. p. 3. ISBN 3-540-67457-8.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. p. 82-84. ISBN 0-19-853952-5.
- ^ Sudbury, Anthony (1986). Quantum mechanics and the particles of nature: An outline for mathematicians (1st ed.). Cambridge University Press. p. 300. ISBN 0-521-27765-5.
- ^ Greiner, Walter (2000). Relativistic Quantum Mechanics: Wave Equations (3rd ed.). Springer. p. 4. ISBN 3-540-67457-8.
- ^ Sudbury, Anthony (1986). Quantum mechanics and the particles of nature: An outline for mathematicians (1st ed.). Cambridge University Press. p. 300-309. ISBN 0-521-27765-5.
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 25,30-31,55-69. ISBN 0-201-62460-5.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. ISBN 0-19-853952-5.
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. ISBN 0-201-62460-5.
- ^ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford Science Publications. p. 55-56. ISBN 0-19-853952-5.
- ^ Kane, Gordon (1994). Modern Elementary Particle Physics: The Fundamental Particles and Forces (Updated ed.). Addison-Wesley Publishing Co. p. 16. ISBN 0-201-62460-5.
- S. Hildebrandt, "Analysis II" (Calculus II), ISBN 3-540-43970-6, 2003
- L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988
- J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley ISBN 0-471-30932-X