Some technical calculations of Klein-Gordon field
Problem
Classical electromagnetism (with no sources) follows from the action
where .
- Derive Maxwell’s equations as the Euler-Lagrange equations of this action, treating the components
as the dynamical variables. Write the equations in standard form by identifying
and
.
- Construct the energy-momentum tensor for this theory. Note that the usual procedure does not result in a symmetric tensor. To remedy that, we can add to
a term of the form
, where
is antisymmetric in its first two indices. Such an object is automatically divergenceless, so
is an equally good energy-momentum tensor with the same globally conserved energy and momentum. Show that this construction, with
leads to the an energy-momentum tensor
that is symmetric and yields the standard formula for the electromagnetic energy and momentum densities:
Solution
- The Lagrangian for this no-source classical electromagnetism is
We can make the variation of its action:
Hence we have
By noting that
then we have for
,
and for
,
- From the previous calculation we can write the variation of action as
Since the variation of Lagrangian is
, then we have
such that we can write the energy-momentum tensor as
We can write the new (symmetric) energy-momentum tensor by adding the new term to the old one:
Obviously
is symmetric. We can find the value of the components of this energy-momentum tensor. The energy density of the electromagnetic field is
And the Poynting vector is given by
Problem
Consider the field theory of a complex-valued scalar field obeying the Klein-Gordon equation. The action of this theory is
It is easiest to analyze this theory by considering and
, rather than the real and imaginary parts of
, as the basic dynamical variables.
- Find the conjugate momenta to
and
and the canonical commutation relations. Show that the Hamiltonian is
Compute the Heisenberg equation of motion for
and show that it is indeed the Klein-Gordon equation.
- Diagonalize
by introducing creation and annihilation operators. Show that the theory contains two sets of particles of mass
.
- Rewrite the conserved charge
in terms of creation and annihilation operators, and evaluate the charge of the particles of each type.
- Consider the case of two complex Klein-Gordon fields with the same mass. Label the fields as
, where
. Show that there are now four conserved charges, one given by the generalization of part 3, and the other three given by
where
are the Pauli sigma matrices. Show that these three charges have the commutation relations of angular momentum
.
Solution
- We start with our Lagrangian density
then we can find the conjugate momenta, i.e.
We can introduce the operators
and
such that
If
, then the field is real. We also can find the expression for the conjugate canonical momenta
Let’s check the commutation relation between
and
.
It shows us that the measurement of position and momentum of this particle in two distinct points of spacetime is independent each other; in every point of spacetime, we cannot measure these two simultaneously. The same calculation will work for
and
, i.e.
. But it is not the case for
and
.
And the last relation maybe is the most simple,
.
- The Hamiltonian density
is expressed as
- We just do some straightforward calculations in this subproblem,
- The charge operator is defined as
Then we can do this calculation
- Now we have two complex-valued Klein-Gordon fields
and
. Define
then we can form the Lagrangian for this fields
We can find the current density of this Lagrangian under this invariant transformations
such that
By inserting the variations of each fields, we will have the current and charge densities,
Next we make a new convention for the symbols
and
, by overriding the previous one, i.e.
By tuning the values of
and
, we have these charges
with
is the Pauli sigma matrices. We can conclude that
where
. Hence we can see easily that
obeys the commutation relations of
group. Namely,
[Useless] theory for logical and lovical behaviors
Few weeks ago my friend and I made a theory that describes our logical and lovical behaviors. It’s disgusting.
We, the human, or more specifically, the nerd human being, have generally two states that represent our mind and heart states. The fields who represent them are called
, the state of logic, and
, the state of love. If we are known to be in the logic (love) state, then at that time our mind (heart) works maximally without being interfered by the others. Since ideally in our daily life we cannot be in one of these states perfectly (suppose you are in a full concentration to read a paper or book in campus park, but suddenly there is a beautiful girl walks beside you), then our state is the superposition of both of them:
where the state
also contains the state
which comes from the fact that we cannot always categorize our behaviors as logical or lovical, but we hope that
. After doing some experiments, it is the result:
,
, and
. Well, at least we must keep
not to be zero identically. However, what is the meaning of the large value of
? It’s simple. Suppose you are in a full concentration to read a paper or book in campus park, but suddenly there is a beautiful girl walks beside you, and indeed she comes to you, also brings many papers and books which are about her homework, and asks you with the dangerous eyes to work that problem. And we cannot say any words while solving it out. And finally the story is ended with a thank you and she walks away, leaves us alone in a park, and we continue to read the damn paper/book.
Well, perhaps we should throw any calculation about this stuff, before everything goes to useless.
The very brief review of Lagrangian and Hamiltonian field theory
— 1. Lagrangian and Hamiltonian Theory —
— 1.1. Lagrangian Field Theory —
Suppose we have a system with the Lagrangian . Then the actual process that can happen is the one where the action
is minimum, i.e. for
then we have
such that we have the equation of motion
As an example, if we have the Lagrangian
then the equation of motion resulted from the Lagrangian above is the Klein-Gordon equation
Note that the field in this equation is not the quantum-mechanical wavefunction in nonrelativistic quantum mechanics, but the classical field instead.
— 1.2. Hamiltonian Field Theory —
For the discrete system, the canonical coordinate is denoted as and its canonical momentum is
, such that the Hamiltonian of this system is
But if we work in the continuous system where we must work with the field , then its canonical momentum is
such that the Hamiltonian of this system can be expressed as
For the Lagrangian , we will have the Hamiltonian
— 1.3. Noether Theorem —
There is a close link between the concept of symmetry of the Lagrangian and the conservation law that is observed in nature. If we change the field such that it will have the form
such that the Lagrangian will be of the form
The relation between and
, then, is
such that we can conclude that this equation holds
An important conserved variable for this current is the charge,
such that
The infinitesimal change of field can also be viewed as the transformation of the coordinate in the spacetime. As an example, the coordinate transformation
corresponds with the change of field
such that the Lagrangian also changes like
,
and hence we can write
where the tensor is called the energy-momentum tensor.
The Hamiltonian density then can be described as the zero-zero component of this energy-momentum tensor, i.e.
and the momentum is
Riemannian Manifolds – Part 2
— 1.4. Torsion and Curvature Tensors —
We have talked so far about the properties which differ the curved manifold with the flat one, such as whether the connection coefficient vanishes or not, but we still don’t have the intrinsic mechanicsm which allows us to determine the curvature of manifold, in the sense that this mechanism must not rely on the coordinate system heavily. The nonvanishing connection coefficient cannot provide these requirements for us for two reasons. First, because it depends on how we make the system of coordinate basis vectors over the manifold and it ends up with how we choose the coordinate system. And second, because it cannot tell us how much the curvature of some region of a manifold. In this subsection we will discuss about the properties of torsion tensor and some curvature tensors. We also discuss the torsion tensor here because its geometrical meaning is as important as the curvature tensors. Here, we will talk about torsion tensor first before attacking the curvature tensors because of the former is simpler than the latter.
The torsion tensor is defined as
for the vector fields . Componentwise, it is described as
such that we have
Therefore, torsion tensor measures how much we differ from the symmetric property of connection coefficient in the Levi-Civita case. If , then the manifold is called torsionless manifold.
If we have three points with their coordinates
,
and
respectively, with
for all
. If we construct the vectors
and
such that
(
) is the vector which connects the point
to
(
). If we parallel transport the vector
(
) along the infinitesimal line
(
), then the component of vector
(
) becomes
(
). This transported vector of
(
) connects a point
(
) to a new point
(
), such that we have the vector which connects points
and
(
and
) is
(
), i.e.
The difference between the vectors and
is
where we notice that the term inside the parenthesis is the torsion tensor . Hence, if the torsion tensor vanishes, then the parallelogram made of small displacement vectors and their parallel transport is closed.
The first curvature tensor I want to discuss is the Riemann curvature tensor . It is defined as
for . The component of Riemann tensor is described as
such that we have
The geometrical meaning of Riemann tensor is about the difference of two vectors after they are parallel transported from the same vector along two different paths. Suppose we have four points which have coordinates
,
,
and
respectively. If we have a vector
, then after parallel transport to point
along the path
, this vector has the form
and similarly for the path ,
where we omit the terms larger than second order in and
. The difference between these two vectors is
and again, we can identify the term in parenthesis as the Riemann tensor .
The Ricci curvature tensor is defined as
and its component is
such that we can form the Ricci scalar curvature, defined as the contraction of Ricci curvature tensor with the metric,
Feynman plays bongo
I know that Feynman’s hobby is to play bongo (and I also see his photo in his Feynman Lectures book), but this video tells me about his skill.
This is the last day, but the beginning of new adventure; Gell-Mann Conference – Day 3
Yesterday after the conference was finished, all participants went to Regent Hotel to attend the banquet. Well I’m not interested to tell something about the banquet, because it’s typical. Moreover, I just took photo of the front of NEC that evening, and took no photo in the banquet party, which means it’s not my enjoyment to come to this type of event.
The conference today is very exciting, especially in the debates from Dvali and others, which make the situation alive. This is the list of speakers, complete with their titles of presentations:
Stephen Adler, from IAS, Princeton University. Dark Matter Scattering and the Flyby Anomalies.
Georgi Dvali, from New York University. Microscopic Gravity and Particle Physics.
Paul Frampton, from University of North Carolina. Primordial Black Holes as All Dark Matter.
Viatcheslav Mukhanov, from Ludwig-Maximilians University. Quantum Origin of the Universe Structure: Theory and Observations.
Crazy!; Gell-Mann Conference – Day 2
This is my report about what happened yesterday. If you ask it about how it is, I’ll say it’s crazy.
The conference was started by the talk from Hagen Kleinert, Free University of Berlin, about Gauge Structures in Multivalued Fields. He had clear ways to explain something, and in fact, he also made a book about Path Integral and Multivalued Fields which are sold here in front of the Auditorium by the World Scientific. The next speaker is Rabindra Mohapatra from University of Maryland who talked about Neutrino Masses and Grand Unification of Flavor. The last speaker in this first session of the day before tea break is Serguey Petcov from SISSA, Italy, and his presentation titled Neutrino Mixing, Oscillations, Leptonic-CP Violation, the See-Saw Mechanism and Beyond.
Session 2 after tea break was started by a presentation about Some Recent Progress in AdS/CFT by John Schwarz from Caltech. I think I enjoy this talk very much. Then, it is followed by the presentations by Ignatios Antoniadis from CERN, Dieter Lust from Ludwig-Maximilians University, Lars Brink from Chalmers University of Technology, Sweden, and Spenta Wadia from Tata Institute, India, with the topics respectively, Aspects of String Phenomenology, String Corrections to QCD, Maximal Supersymmetry and Exceptional Groups, and Gauge/Gravity Duality and Some Applications.
It is a really great day; Gell-Mann Conference – Day 1
And the first day of Gell-Mann conference finally finished tonight. There were so many interesting things happened today, and those all started with a kind of not-so-delicious-because-I’m-not-so-hungry breakfast this morning, yet because I had eaten too much that night.
Anyway, the conference was started at 8 a.m. this morning, begun by the opening talk of Su Guaning, the President of NTU. Following a speech from Bertil Andersson (provost of NTU), there were some sequential talks from Francis Yeoh from the National Research Foundation, Singapore, whose topic Strengthening Research. Growing Innovation and Enterprise, Teck Seng Low from Agency for Science, Technology and Research, Singapore, with a topic The Nexus of Science: Key to Spawning New Innovations and Discoveries, and Choy Heng Lai from NUS, where his talk is about Overview of Complexity and Quantum Information Research in Singapore. These talks are mainly about the research situation in Singapore, not much the topics I want to listen in this conference.
The next speaker is Harald Fritzch from Ludwig-Maximilians University. He talked about Murray Gell-Mann – A Scientific Biography, but actually it was not only about the biography; the Gell-Mann’s idea about quarks concept is also explained. After that the physicist from MIT, George Zweig, also gave a talk titled Memories of Murray and the Quark Model. Generally the Zweig’s talk is more detail than Harald’s about the concept of quark revealed by Gell-Mann, but Harald Fritzch’s talk is broader, in the sense that he also told us about the personal life of Gell-Mann; something that is quite rare in this first day of the conference.
After this first session there was a tea break. Well, it’s the part that I cannot tell too much. The whole story is about eating and eating (and drinking, of course).
After the tea break, the first speaker is the Nobel laureate, Chen Ning-Yang, from Tsinghua University (or Brookhaven?). He talked much about Some Problems in Cold Atom Research. Well, rumor (ya, this rumor is from my friends in NTU, they heard it from the Yang’s talk 2 years ago) said that when Yang was 72 years old, his wife was 27. The first time I heard it, besides counting the difference of these numbers, was thinking about the breaking of this permutability after the time goes by; well don’t take it too seriously. Next is Nicholas Samios from Brookhaven National Laboratory, who talked about Murray and . And the next two speakers are Marek Karliner from Tel-Aviv University, Israel, and Gabriel Karl from University of Guelph, who talked about From
to
and Early History of QCD and Quarks, respectively.
And finally I arrived in Singapore; Gell-Mann Conference – Day 0
Yak, here’s the thing. I just arrived in Nanyang Executive Center (NEC), one of convenient place here in Nanyang Technological University (NTU), where since the last 4 years I have visited this place twice; the first is when I took part in IPhO XXXVII, July 2006. And of course my arrival at this time has a strong reason. You can assume that I take a holiday in Singapore, but it’s actually not true, because if I wanna enjoy my holiday, I won’t go to NTU.
I am here to attend the Conference in Honour of Murray Gell-Mann’s 80th Birthday, with the topics are all about Quantum Mechanics, Elementary Particles, Quantum Cosmology and Complexity. I can say that this is a great conference held near my university, because it has many notable speakers in the main session. I see many physicists outside there but I still cannot recognize who they are. Just be patient until they present their talks tomorrow.
Above is one of corners in my room. Well, I try to write shortly about my route to arrive in this place. When I landed in Changi Airport, my position was in Terminal 1. After taking the free-of-charge Skytrain to Terminal 2, I continue walking to the MRT (Mass Rapid Transfer) station in the basement of Terminal 2. There, I bought a ticket to the Boon Lay MRT station, with a transfer in Tanah Merah. In Boon Lay, I walked to the nearest BUS station, in a direction for NTU and NIE. I took BUS 179 and paid it using coins, S$1.20. Then I stopped in B05 Canteen 2, a bus stop inside NTU. After 5-minutes walking finally I arrived here in NEC. Well, the real condition was not so straight and easy like I describe above. There were too many distractions, such as asking someone or losing direction, which surely made me pain and tired. But overall, this place is kinda easy to find.
Tomorrow will be a fantastic day, and of course it will be energy-consuming. Now I want to take a nap.
Riemannian Manifolds – Part 1
— 1. Riemannian Manifolds —
— 1.1. Metric Tensor —
Now it’s the time to introduce the concept of distance into our manifold. In the previous posts, we have defined the curved space and generalized some objects which initially only exist in the flat space such as vector and dual vector fields, and regard them as the quantities which have the representations in terms of coordinate basis and hence also have the components, relative to this coordinate. However, in daily life we must tackle the space which has a distance concept within it and the concept of length of vector. The latter has a great significant functionality because often in physics we need to know the magnitude of quantity defined as the vector and dual vector fields. The development of the topics discussed in this post is useful for the construction and formulation of general relativity.
For if we want to define a distance notion in a given manifold , then we can simply take any two arbitrary points in
and define how much the distance between them. Of course, we can say that our manifold has a distance notion only after the distance of any pairs of its points has been defined. Notice that this distance is not affected by our choice of coordinate; the type of coordinate system and the way we use it won’t change the distance we have defined between pairs of points. Hence, we should construct a mechanism which allows us to pretend the distance of two fixed points although we change the coordinate system of our manifold.
If we have two nearby points which have the coordinate values
and
, then the distance between them should be comparable to the difference between their coordinates, i.e.
, for
. Precisely, if we make a vector
which connects
and
, then we can represent it in terms of basis vectors as
where is the basis vector, such that the distance
between these points can be stated as the length of this vector, i.e.
If we define
then we can have the convenient description for the distance of points and
as
Since is the inner product of two basis vectors, we can interpret it as the component of a
-tensor
where for the next discussion will be called as the metric tensor, and we can write it in terms of basis of the vector space
,
where now the is not the difference between coordinates of
and
which most physicists will imagine it as the infinitesimal distance, but it should be considered as the basis of dual vectors in
. Being a
-tensor, it needs two vectors to produce a real number, and if we provide the vector
for its argument we will get the equation of
above. Since
and
can be interchanged, then we need the metric
to be symmetric in its indices,
If we have two vectors , then the inner product between them is
such that if we set , then we have the length of vector
,
But it still doesn’t tell us the whole story. We can restrict the metric to be positive-definite, in the sense that if we have a vector
, then the norm of this vector
must be nonnegative, or
, where the equation holds only for the case of the nullity of
. If a manifold
is equipped with the metric
which is symmetric and positive-definite, then this manifold is called the Riemannian manifold, and its metric
is called the Riemannian metric. Since it is positive-definite, all eigenvalues of Riemannian metric are positive, and hence it has the inverse denoted as
. Moreover, it also can be diagonalized by using certain procedure involving the orthogonal matrix, and rescaled such that all entries in the diagonal of diagonalized metric are the unity. This matrix is nothing other than the identity metric, which is also known as the metric tensor of the Euclidean space
, denoted as
. Therefore, we conclude that the Riemannian metric
of a Riemannian manifold
can be reduced to the identity metric
of the Euclidean space
.
We can weaken our restriction about the positive-definiteness of metric to a more flexible condition: can be zero although
is not zero, and it is possible to have
for a certain
. If a manifold
is equipped with this kind of symmetric metric, then
is called pseudo-Riemannian manifold, and its metric
is called pseudo-Riemannian metric. We can principally reduce this metric into the diagonal matrix, and hence we will get some positive and negative eigenvalues, or after rescaling and reordering, we will have the diagonal matrix
, in which there are
numbers of
and
numbers of
. The signature of metric is defined as
, and it is same for all points in the manifold. For the specific case, if the number of positive eigenvalues is only one, then
is called the Lorentzian manifold, and its metric
is called the Lorentzian metric. The diagonal metric corresponding to the Lorentzian one is called the Minkowski metric,
.
Let’s back to the case of Riemannian manifold with a Riemannian metric
. If this metric is provided only with one vector
, i.e.
, then it will produce something like the dual vector, since this object needs one more vector to produce a real number. Therefore, the metric
can be a bridge between the set of vectors and dual vectors in the manifold. If we denote the dual vector corresponding to
as
, then
such that componentwise it has the form
If the vector is taken into the empty slot of
, then we have
where the operation of inner product now can be carried out easily by using this new tool.
Here is the clear ground for this matter. In the manifold where we don’t introduce the metric, the inner product takes place between a vector and a dual vector. And since there is no link between the set of tangential vectors and dual vectors
, we can take a product between any vectors and any dual vectors without an interesting intuitive sense. If we are working in the (pseudo-)Riemannian manifold, then the inner product also takes place between a vector and a dual vector. But since there is a linking chain between them, so called as the metric, then we will have an interesting situation when we take an inner product between a vector and its corresponding dual, in which it will give us the norm or length of that vector (or, dual vector). Of course we can freely take an inner product between a vector with any dual vectors which don’t correspond to that vector, in which we will have the ordinary inner product between two different vectors.



























