Sunday, 23 May 2010

The Moon

This is the best image from fifteen photographs of the Moon that I took this evening. My flat has a skylight thing in the roof, which opens and tilts in various ways (it would be great for using a telescope, actually...) so I decided to set up my camera and tripod and take some photos of the moon, since it was a clear night.

Thursday, 4 March 2010

The Road To Quantum Field Theory: Updated Post List

Once again, I'm posting an index, essentially, of the posts in the `Road to QFT' series, so far. The previous list was here, but I'll reproduce all of the links in this post too.

Pre-Series Posts
These were posts on related topics (special relativity, quantum mechanics and particle physics)
The `Road To Quantum Field Theory' Series
Posts in the `Road to QFT' series, in chronological order.
Out-of-Series Posts
Posts on related topics which were not intended as part of the `Road to QFT' series, but may be of interest anyway.
Roadmap
As before, I'll post a rough list of things still to be covered. By comparing the previous roadmap to the most recent two posts, you'll notice I've done things slightly out of the order I mentioned there, so this is really a very approximate guide.
  • Relativistic QM: The Klein-Gordon Equation & Spin-Zero Particles
  • Relativistic QM 2: The Dirac Equation & Spin-Half Particles
  • Quantum Electrodynamics (QED)
  • Quantum Chromodynamics (QCD)
  • SU(2) and Electroweak Unification
  • Quark Flavour Mixing
  • Spontaneous Symmetry Breaking and the Higgs Mechanism
  • Beyond the Standard Model: Neutrino Mass Terms & Neutrino Mixing
This is still a very ambitious set of things to cover. In particular, the posts so far have only brushed the surface, providing the background needed to understand relativistic quantum mechanics and gauge theories. The real work is yet to come! Note also that I am not a theorist, so my coverage of some of the more advanced topics may be less than completely thorough! I've added sections on quark flavour mixing and neutrino masses & mixing to the roadmap since they are of particular interest to me.

    Tuesday, 2 March 2010

    Gauge Invariance in Quantum Mechanics

    This post continues the 'Road to Quantum Field Theory' series.

    The Lorentz force law for a non-relativistic particle of charge q, moving with velocity v in both electric (E) and magnetic (B) fields, is given by:
    \mathbf{F} = q\mathbf{E} + q\mathbf{v}\times\mathbf{B}
    This can be derived from Hamilton's equations using the classical Hamiltonian given below.
    H = \frac{1}{2m}\left(\mathbf{p}-q\mathbf{A}\right)^2 + qV
    The Schrödinger equation for a charged particle in an EM field is,
    \left[\frac{1}{2m}\left(-i\nabla - q\mathbf{A}\right)^2 + qV \right] \psi(\mathbf{x},t) = i\frac{\partial \psi(\mathbf{x},t)}{\partial t}
    obtained from the Hamiltonian through the substitution
    \mathbf{p} \rightarrow -i\nabla
    as is usual for the quantum mechanical momentum operator. We can identify operator combinations:
    \mathbf{D} = \nabla - iq\mathbf{A}
    D^0 = \frac{\partial}{\partial t} + iqV
    which replace the operators
    \nabla, \frac{\partial}{\partial t}
    when we move from the free-particle Schrödinger equation to the electromagnetic field case.

    The solutions for the wavefunctions of the Schrödinger equation describe completely the behaviour of a particle under the influence of the potentials A and V, but these potentials are not unique, as I showed in the previous post in this series. Instead, they can be changed by a gauge transformation:
    \mathbf{A}\rightarrow\mathbf{A}^\prime=\mathbf{A}+\nabla\chi
    V\rightarrow V^\prime = V - \frac{\partial\chi}{\partial t}
    Maxwell's equations for E and B will remain invariant under these transformations (this was the topic of the previous post), but will the physics described by the Schrödinger equation be the same if we make these changes to the potentials there?

    The answer is no! The Schrödinger equation is not gauge invariant, since the same wavefunction cannot satisfy both the original and transformed versions! All is not lost, however. The wavefunction itself is not directly observable, whereas E and B are. If we do not require the wavefunction to remain invariant when the potentials undergo a transformation, then we are free to change the wavefunction such that we retain invariance of physical observables. Write the Schrödinger equation in terms of some transformed wavefunction,
    \psi\rightarrow\psi^\prime
    i.e.
    \left[\frac{1}{2m}\left(-i\nabla - q\mathbf{A}^\prime\right)^2 + qV^\prime\right] \psi^\prime(\mathbf{x},t) = i\frac{\partial \psi^\prime (\mathbf{x},t)}{\partial t}

    Now, the form in which we have written the equation above is identical to that of the original Schrödinger equation, except that we are writing it in terms of primed quantities (psi', A', V') instead of unprimed (psi, A, V). Both equations describe the same physics, so if we can find such a psi', then the Schrödinger equation is gauge covariant; that is, it maintains the same form under a gauge transformation.

    We know the relationship between A, V and A', V', so we can write down the transformation for psi':
    \psi^\prime (\mathbf{x},t) = e^{iq\chi(\mathbf{x},t)}\psi(\mathbf{x},t)
    where
    \chi
    is the same space- and time- dependent function appearing in the transformations of A and V. We can verify that this results in the gauge covariance of the Schrödinger equation:
    (-i\nabla - q\mathbf{A}^\prime)\psi^\prime=\left[-i\nabla-q\mathbf{A}-q(\nabla\chi)\right]e^{iq\chi}\psi
    =q(\nabla\chi)e^{iq\chi}\psi + e^{iq\chi}(-i\nabla\psi) + e^{iq\chi}(-q\mathbf{A}\psi) - q(\nabla\chi)e^{iq\chi}\psi
    The first and last terms cancel, leaving:
    (-i\nabla - q\mathbf{A}^\prime)\psi^\prime = e^{iq\chi}(-i\nabla - qA)\psi
    which can be written as:
    (-i\mathbf{D}^\prime \psi^\prime) = e^{iq\chi}(-i\mathbf{D}\psi)


    The space-time dependent phase factor feels the action of
    \nabla
    , but passes through the combined D' operator, converting it to D, so it becomes clear that D'psi' is related to Dpsi in the same way psi' is related to psi! Similarly,
    (iD^{0\prime}\psi^\prime = e^{iq\chi}(iD^0\psi)

    We can now erite:
    \frac{1}{2m}(-i\mathbf{D}^\prime)^2 \psi\prime = e^{iq\chi}\frac{1}{2m}(-i\mathbf{D})^2 \psi = e^{iq\chi}iD^0\psi = iD^{0\prime}\psi^\prime
    which demonstrates the correct relationship between psi and psi'. The question remains whether the same physics is described by both psi and psi'. We can see that it is in a number of ways:

    1. The probability density is given by
    |\psi|^2 = \psi^\dagger \psi
    , which is equivalent to
    |\psi\prime|^2 = \psi^{\prime\dagger} \psi^\prime = \psi^\dagger e^{-iq\chi}e^{+iq\chi}\psi = \psi^\dagger \psi

    2. The probability current,
    \psi^\dagger(\nabla\psi) - (\nabla\psi)\psi^\dagger
    is not invariant under a gauge transformation, but when we replace
    \nabla \rightarrow \mathbf{D} ~~,~~ \frac{\partial}{\partial t} \rightarrow D^0
    then,
    \psi^\dagger\prime(\mathbf{D}\psi^\prime = \psi^\dagger e^{-iq\chi} e^{iq\chi}(\mathbf{D}\psi) = \psi^\dagger\mathbf{D}\psi
    (and similarly for the other term).

    Hence, identical physics is described by both the wavefunction psi, and the gauge transformed wavefunction psi'; the gauge invariance of Maxwell's equations presents as a gauge covariance in quantum mechanics, provided that we transform not only the potentials, but the wavefunction also:
    \mathbf{A}\rightarrow \mathbf{A}^\prime = \mathbf{A} + \nabla\chi
    V\rightarrow V^\prime = V - \frac{\partial \chi}{\partial x}
    \psi\rightarrow \psi^\prime = i^{iq\chi}\psi

    Finally, we note that the new differential operators,
    \mathbf{D} = \nabla - iq\mathbf{A} ~~,~~D^0 = \frac{\partial}{\partial t} + iqV
    can be written in a Lorentz covariant form:
    D^\mu = \partial^\mu + iqA^\mu

    This allows us to write:
    -iD^{\prime\mu}\psi^\prime = e^{iq\chi}(-iD^\mu \psi)
    and it follows that an equation can involving the operator
    \partial^\mu
    can be made to be gauge invariant by under the combined transformations:
    A^\mu \rightarrow A^{\prime\mu} = A^\mu - \partial^\mu \chi
    \psi \rightarrow \psi^\prime = e^{iq\chi}\psi
    provided the minimal substitution is also made:
    \partial^\mu \rightarrow D^\mu = \partial^\mu + iqA^\mu

    This provides a simple mechanism to get the wave equation for a particle in an electromagnetic field from the equation for a free particle - by making the above substitution. This forms the basis of the gauge principle, i.e. the form of an interaction is determined by an insistence on local gauge invariance.

    That's enough for now; next time we'll move on to study the Klein-Gordon equation for relativistic spin-zero particles.

    Monday, 8 February 2010

    Farscape Intro Randomiser

    While watching Farscape Season 2, Episode 4 (Crackers Don't Matter) last night, I wrote a Python script to randomise elements of the Farscape intro. Here's an example:

    My name is John Crichton, a wormhole hit and I got shot through a military commander. Now I'm lost in some distant part of the universe on an insane radiation wave, full of escaped prisoners. I'm being hunted by a living ship. Doing everything I can. I'm just looking for a way home.



    import random
    
    adjectives = [
     ('insane', 'an'),
     ('living', 'a'),
     ('escaped', 'an'),
     ]
    
    nouns = [
     ('ship', 'a'),
     ('radiation wave', 'a'),
     ('military commander', 'a'),
     ('prisoner', 'a'),
     ('wormhole', 'a')
     ]
    
    
    parts = []
    parts.append(random.choice(nouns))
    nouns.remove(parts[0])
    parts.append(random.choice(nouns))
    nouns.remove(parts[1])
    parts.append(random.choice(adjectives))
    adjectives.remove(parts[2])
    parts.append(random.choice(nouns))
    nouns.remove(parts[3])
    parts.append(random.choice(adjectives))
    adjectives.remove(parts[4])
    parts.append(random.choice(nouns))
    nouns.remove(parts[5])
    parts.append(random.choice(adjectives))
    adjectives.remove(parts[6])
    parts.append(random.choice(nouns))
    nouns.remove(parts[7])
    
    result = 'My name is John Crichton, '
    result += parts[0][1] + ' ' + parts[0][0]
    result += " hit and I got shot through "
    result += parts[1][1] + ' ' + parts[1][0]
    result += ". Now I'm lost in some distant part of the universe on "
    result += parts[2][1] + ' ' + parts[2][0] + ' ' 
    result += parts[3][0]
    result += ", full of "
    result += parts[4][0] + ' ' + parts[5][0] + 's'
    result += ". I'm being hunted by "
    result += parts[6][1] + ' ' + parts[6][0] + ' ' + parts[7][0]
    result += ". Doing everything I can. I'm just looking for a way home."
    
    print result
    

    Tuesday, 8 December 2009

    Lorentz-Covariant and Gauge-Invariant Electrodynamics

    Over a year ago I started a series of blog posts aimed at describing quantum field theory, beginning with Lagrangian mechanics and working all the way up. It's been a while since I made any posts in that series. Indeed, the last post was a roadmap of sorts; a list of the topics still to be covered.

    The roadmap hinted that the next step is a relativistic description of electromagnetism. Actually, what we really want to do is describe electromagnetism as a gauge theory. That is, a theory which is gauge invariant. By the end of this post, what we mean by gauge invariance will become obvious.

    So, here goes.

    Maxwell's Equations
    Any treatment of electromagnetism must start with Maxwell's equations, presented below.

    (1)\,\,\,\,\,\nabla\cdot\mathbf{E} = \rho_{em}

    (2)\,\,\,\,\,\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}

    (3)\,\,\,\,\,\nabla\cdot\mathbf{B} = 0

    (4)\,\,\,\,\,\nabla\times\mathbf{B} = \mathbf{j}_{em}


    Here,
    \rho_{em}
    and
    \mathbf{j}_{em}
    are the electric charge density and electric current density, respectively, and we are working in Heaviside-Lorentz units. The densities act as the sources of the electric and magnetic fields.

    Taking the divergence of (4) leads to a problem; the continuity equation for electric charge states that
    (5)\,\,\,\,\,\frac{\partial \rho}{\partial t} + \nabla\cdot\mathbf{j} = 0

    (note that I will omit the "em" subscript on the densities from now on).

    Since
    \nabla\cdot\nabla\times\mathbf{B} = 0\Rightarrow\nabla\cdot\mathbf{j}=0

    This is only true when the charge density does not vary with time. In general, Ampere's law must be modified to be
    (6)\,\,\,\,\,\nabla\times\mathbf{B}=\mathbf{j}+\frac{\partial\mathbf{E}}{\partial t}

    This is now consistent with (5), the continuity equation.

    The continuity equation states that the rate of change of charge in some volume is due entirely to the flux of current through its surface. That is to say, electric charge is conserved. Since this volume can be made arbitrarily small, we can say that electric charge is locally conserved. This means that we can't balance things out by having a negative charge appear out of nowhere, as long as a positive charge is made the other side of the universe (or a negative charge disappears on the other side of the universe); this conservation law has to apply on an arbitrarily small scale.

    We can introduce the vector potential
    A_\mu
    in place of the fields E and B:
    (7)\,\,\,\,\,\mathbf{B}=\nabla\times\mathbf{A}

    (8)\,\,\,\,\,\mathbf{E}=-\nabla V-\frac{\partial \mathbf{A}}{\partial t}


    This defines the three-vector potential A, and the scalar potential V (sometimes called the electrostatic potential, and sometimes denoted
    \phi
    . Equations (2) and (3) are then satisified automatically.

    The gauge invariance of electromagnetism is based on the fact that, for a given E and B, the equations above do not uniquely determine A and V. In fact, one can apply transformations to A and V which leave E and B completely unchanged; and therefore leave all of the physics unchanged. These are known as gauge transformations, and the invariance of Maxwell's equations under these transformations is known as gauge invariance

    We are free to change A as follows;
    (9)\,\,\,\,\,\mathbf{A}\rightarrow \mathbf{A}^\prime = \mathbf{A}+\nabla\chi

    where
    \chi
    is an arbitrary scalar function. Since
    \nabla\times(\nabla\chi) = 0
    , this does not change B.

    To preserve E when making the above transformation, we must simultaneously transform the scalar potential V:
    (10)\,\,\,\,\,V\rightarrow V^\prime=V-\frac{\partial\chi}{\partial t}


    The transformations (9) and (10) can be written in a single expression using the four-potential,
    (11)\,\,\,\,\,A^\mu = (V, \mathbf{A})


    Note now that the differential operators,
    (12)\,\,\,\,\,\left(\frac{\partial}{\partial t}, -\nabla\right)

    form a four-vector operator
    \partial^\mu
    . The gauge transformation can then be specified by:
    (13)\,\,\,\,\,A^\mu \rightarrow A^{\prime\mu} = A^\mu - d^\mu\chi


    Maxwell's equations can be written in a manifestly Lorentz covariant form (meaning, they have the same form under Lorentz transformations) using the four-current:
    (14)\,\,\,\,\,\mathbf{j}^\mu = (\rho,\mathbf{j})


    The continuity equation can then be written:
    (15)\,\,\,\,\,\partial_\mu j^\mu = 0

    which in turn allows us to write equations (1) and (6) as:
    (16)\,\,\,\,\,\partial_\mu F^{\mu\nu} = j^\nu


    The equation above introduced the Field Strength tensor (or the Faraday tensor),
    (17)\,\,\,\,\,F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu


    Under a gauge transformation (13), the Faraday tensor (17) remains unchanged; it is gauge invariant, and so, therefore, are Maxwell's equations in the form (16). As I already said above, Maxwell's equations are Lorentz covariant in the form (16), leading to a Lorentz covariant and Gauge invariant theory.

    Now, one can write
    (18)\,\,\,\,\,\Box A^\nu - \partial^\nu(\partial_\mu A^\mu) = j^\nu


    Equation (6) can be inferred (and indeed, was inferred by Maxwell) from the local charge conservation requirement expressed by the continuity equation (15). The field equations (16) automatically include the continuity equation. Mathematically speaking, the Faraday tensor (17) is a sort of "four-dimensional curl". This is unchanged by the gauge transformation (13), suggesting that gauge invariance in electromagnetism is related to charge conservation. The reality is a little more complicated.

    It was shown in 1949 by Wigner that the principle that no physical quantity can depend on the absolute value of the electrostatic potential V, combined with conservation of energy, implies that charge conservation holds. This relates charge and energy conservation to an invariance under the transformation of the electrostatic potential by a constant. Charge conservation alone does not require the more general space-time dependent transformation of the gauge invariance we've discussed above.

    Changing the electrostatic potential by a constant amount is an example of a global transformation. Invariance under this global transformation is related to the conservation of electric charge, but it is not sufficient to obtain all of electromagnetism. Instead, we must impose a local change in the electrostatic potential V (the time-derivative term in (10), which is compensated by a simultaneous change in the vector potential A, leaving Maxwell's equations ultimately unchanged. By including these magnetic effects, the global invariance under a change in V, related to conservation of electric charge, can be extended to a local invariance.

    The concept of local gauge invariance is important for the development of quantum field theory in subsequent posts. Next time, a tour of gauge invariance in quantum mechanics, and the gauge principle.

    Much of this post was based on Chapter 3 of Aitchison & Hey (Gauge Theories in Particle Physics, 3rd Edition, Vol. I - From Relativistic Quantum Mechanics to QED). It is a book I strongly recommend purchasing if you have an interest in quantum field theory. The second volume discusses QCD and Electroweak theory, and is another good purchase.

    Wednesday, 15 July 2009

    Reversing Lines In A File

    As is often the case, an interesting little text manipulation task came up in the office today; given a file containing lines of text, reverse it (i.e. put the last line first, the penultimate line second, etc.)

    I have a feeling there's a neat little command-line utility that already does this (but can't remember what it is...), but my mind soon went to a short Python script. Here's my first stab at this:

    #!/usr/bin/env python
    import sys

    lines = []
    for line in open(sys.argv[1]):
    lines.append(line.strip())

    lines.reverse()
    for line in lines:
    print line


    After this, I tried to shorten it a bit. Putting the lines into a list could be done nicely with a list comprehension,

    #!/usr/bin/env python
    import sys

    lines = [line.strip() for line in open(sys.argv[1])]

    lines.reverse()
    for line in lines:
    print line


    Finally, I discovered the reversed() function, which allows you to create reverse iterators for any Python sequence. Here's where the list comprehension and Python's iteration stuff really comes into its own, leading to a wonderful two-liner (not counting the module import and the #!)


    #!/usr/bin/env python
    import sys
    for line in reversed([line.strip() for line in open(sys.argv[1])]):
    print line

    Thursday, 19 February 2009

    Writing Scientific Documents Using LaTeX: Updated

    I submitted a fifth edition (5/E) of `Writing Scientific Documents Using LaTeX` to CTAN yesterday (and another version today which fixes a couple of minor mistakes in the examples given).

    As usual, the document and associated files can be found at:
    http://www.ctan.org/tex-archive/info/intro-scientific/