Hidden Dimensions

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Posts Tagged ‘hep_th’

QFT Journal Club 2: Propagator Theory

Posted by Phiphy on 10/08/2009

Time: 10/3/2009

Speaker: James Murray

Outline:

  • Non-relativistic propagators in quantum mechanics.
  • Propagators of scalar field.
  • Propagators of fermion field.
  • The perturbative solution of propagators.

Reference:

Bjorken & Drell, Relativistic Quantum Mechanics, Chapter 6.

Comments:

Only part of the comments were made in the meeting.
  • Propagators as Green’s Functions

The mathematical meaning of propagators are just Green’s functions of the equations of motion (EOM) of the particles or fields.
\widehat{L}(x)G(x,x')=\delta(x-x')
Where \widehat{L}(x) is a linear differential operator.

We learned in mathematical methods of physics that the relationship between Green’s functions and the general solutions of the EOM with source depends on the order of derivatives in the EOM and also on boundary conditions. Generally, if an EOM is second-order in the time derivative, we need to know the initial conditions of both the solution and the first order time derivative of the solution. In non-relativistic quantum mechanics (NRQM), we only have the first order time derivative, so it’s legitimate to write down a general solution of the wave function as
\Psi(x',t')=i\int d^{3}x dt G(x',t';x,t)f(x,t)

This is just an application of Huygens’ principle for evolution of waves. Then in relativistic quantum mechanics (RQM), the Klein-Gorden equation is in the second order of time derivative, but we used the same evolution function for the wave functions. Why? Because we set the initial condition as
\Psi(x,t\to -\infty )=\frac{\mathrm{d} }{\mathrm{d} t}\Psi(x,t\to -\infty )=0
You may notice that there is no boundary terms in space either, because they are also set to be 0.

The Green’s function itself also depends on boundary conditions. It’s remarkable that ‘time ordered’ is just a kind of boundary conditions. In NRQM, we input the step function \theta (t) as a boundary condition that G(t)=0 \forall t<0 . This gives us the retarded Green’s functions. In RQM, the boundary condition becomes F(x,t)=G(x,t) for t>0 and F(x,t)=G(-x, -t) for t<0 , where G(x,t) is the on-shell Green’s function without this time-ordered condition. F(x,t) is just the Feynman propagator.

In RQM or QFT, we only deal with the simplest case with the simplest boundary conditions. As a contrary, in condense matter physics, there may be many weird boundary conditions and the problem becomes much more complicated.

  • Propagators as Correlation Functions

There is another name for propagators which is more commonly used in statistical mechanics: two-point correlation functions. Propagators are just a kind of correlation functions. Since it correlates two states in space-time, it describes the evolution, ie, propagating of the particle or field. In condense matter physics, the more commonly used correlation functions are only defined in pure space, where time is not a variable. Although they have different properties, their essence is the same: for a given system, if we know the probability amplitude of the ‘particle’ at one point in the ’space’, a correlation function gives us the amplitude of the ‘particle’ at another point under such conditions. Here the ‘particle’ can also mean field or abstract states, and ’space’ can also mean space-time or any abstract phase space.

By understanding its physical meaning, it’s not difficult to write down the general and abstract formula for two-point correlation functions:
G(\alpha ,x;\alpha ',x')=\langle \alpha ,x|\alpha ',x' \rangle=\sum_{n} \langle \alpha ,x|n \rangle \langle n|\alpha ',x' \rangle
Where x and x’ are space-time indices and \alpha, \alpha ' are indices for all internal degrees of freedom. |n \rangle are a complete set of states.

  • How physical is a propagator?

In QM, we learned that some quantities are physical and some are not. For example, wave functions are not physical, because they can have different phases for the same physics; but the modulus squared of a wave function or of an inner product of wave functions are physical, because they mean probabilities which can be measured directly. Generally speaking, a physical quantity respect the symmetry of the system, ie, it must be some symmetric representations of the symmetry group, while an unphysical quantity does not necessarily have this constraint.

We’ve seen that propagators are amplitudes, so the question ‘how physical a propagator is’ can be translated into the question ‘ how physical an amplitude is’. In QM, an amplitude is almost physical in the sense that there is only an unphysical phase in the amplitude. So if we impose Lorentz symmetry into the system, it’s natural to require that amplitudes, then also propagators to be Lorentz invariant. However, this is not always the case in RQM. The propagator of a scalar field is Lorentz invariant because the two state correlated are also Lorentz invariant. But we’ll see later that a propagator can be a tensor which is Lorentz covariant, eg, fermion propagator, or it can even have no Lorentz covariance property, eg, photon propagator in Coulomb gauge. This is because each component of the propagator describes the correlation of two components of the field, and even the field components can be unphysical as we’ll see in gauge field theory.

  • Classical and Quantum Propagators

As long as there is a linear differential equation, there is a set of Green’s functions. Physically it means propagators exist in every wave system, no matter it’s quantum or classical. Quantum mechanics (including relativistic quantum mechanics) treats single particles as waves, so it’s also called Wave Mechanics. In this sense, QM is essentially the same as a classical field theory (Of course, QM is a special field theory in the sense that it has only the first order of the time derivative). For example, the Klein-Gorden equation is a quantum equation for relativistic scalar particles, but it’s also a classical equation for relativistic scalar fields. Then what’s the difference between quantum and classical theories? They have different propagators. We use retarded propagators in classical field theories while Feynman (time-ordered) propagators in relativistic quantum mechanics. They have different boundary conditions in time according to different physical conditions. For classical field theory, we do not want any particle or anti-particle to be produced, so we have propagators which start at an initial time point and always propagate forward in time with a positive pole energy. For quantum mechanics, we tried to keep the same property, yet realized that in order to combine quantum mechanics with relativity, we have to end up with a weird propagator in which some negative energy propagates backward in time. Then we have to accept the concept of anti-particles and realized that RQM is in itself inconsistent, because we start with a single particle system yet end up with a system in which particle number is not conserved. The only solution is to further treat particle wave functions as field operators, then we come to quantum field theory. This is called the second quantization. In fact, there is another way leading to QFT, that is to quantize classical field directly, in which all the symmetries and dynamical variables are already prepared.

It is remarkable that the uncertainty principle and the ‘off-shell’ of a propagator are not exclusively the properties of quantum mechanics. In fact they are properties of waves and also exist in classical field theories. As there are sources, the EOM of free waves no longer holds at all time so the dispersion relation changes, and that is the cause of ‘off-shell’.

  • The Role of Propagators in Perturbative Solutions of the EOM

In principle, if we can exactly solve the EOM of a system with a certain boundary condition, all dynamics and hence the state of the system at any future time is determined, then our work is done. However, most of the EOM’s are not easy to be solved analytically, especially when there are interactions. We have seen it in QM that there are only a few exactly solvable systems. So we introduced a powerful tool: perturbation methods. In QFT, solving EOM becomes even more difficult because the EOM with interaction is usually nonlinear and the interaction always depends on time. To use the perturbation method, we have to make two assumptions: first, in order to use the superposition principle of Green’s functions, we assume that the interaction only happens in a finite region of space-time and we are only interested in initial and final states which are far from the interaction region and can be considered as free particles for which the EOM is linear; Second,  we assume that the interaction is small and can be expanded perturbatively, then as in time-dependent perturbation theory in QM, we can calculate the correlation functions iteratively, with different interaction points connected with propagators of free particles (Note: free does not mean on-shell, it just means it’s the propagator solved from the EOM without interactions ).

  • The structure of propagators in QFT

Now we’ve seen two kinds of propagators, the one for scalars:
G(p)=\frac{i}{p^2-m^2+i\varepsilon }
and the one for spinors:
G(p)=\frac{i (\hat{p}+m) }{p^2-m^2+i\varepsilon }
Where \hat{p}=\gamma ^{\mu}p_{\mu} is a matrix carrying spinor indices.
We see that the spinor propagator equals a scalar propagator, which provides the pole and time-order structure, multiplied by some matrix in spinor space. This structure is very general. No matter the field is a scalar, a spinor or a vector field, each component (polarization) of the field propagates like a scalar, then we only need to sum over all the components to get a total propagator, that is the why the numerator of the spinor appears. Let’s use this manner to look at a vector propagator:

A vector field can be written as A_{\mu}(x)={\varepsilon}_{\mu}^{\lambda}a^{\lambda}(x)
Where {\varepsilon}_{\mu}^{\lambda} is the polarization basis and a^{\lambda}(x) is the ‘coordinate’ on this direction which is a pure number. To calculate the propagator, we take a^{\lambda}(x) as a delta function at point x. According to the definition of correlation functions mentioned earlier,
G_{\mu \nu}(x-x')=\left \langle A_{\mu}(x)|A_{\nu}(x') \right \rangle=\left \langle {\varepsilon}_{\mu}^{\lambda}a^{\lambda}(x)|{\varepsilon}_{\nu}^{\lambda'}a^{\lambda'}(x') \right \rangle=\left \langle {\varepsilon}_{\mu}^{\lambda}|{\varepsilon}_{\nu}^{\lambda'} \right \rangle \left \langle a^{\lambda}(x)|a^{\lambda'}(x') \right \rangle
The propagating between polarizations are orthogonal, so in momentum space
\left \langle a^{\lambda}(x)|a^{\lambda'}(x') \right \rangle=\frac{i{\delta}_{\lambda \lambda'}}{p^2-m^2+i\epsilon}
then we have G_{\mu \nu}(p)=\frac{i\sum_{\lambda}{\varepsilon}_{\mu}^{\lambda} {\varepsilon}_{\nu}^{\lambda}}{p^2-m^2+i\epsilon}
This is the the general structure of vector propagators and \sum_{\lambda}\varepsilon_\mu^\lambda \varepsilon_\nu^\lambda is gauge dependent for massless vector field so we see that the propagator is also gauge dependent.

Posted in Communications, Read them | Tagged: | 1 Comment »

QFT Journal Club 1: Groups and Group Representations in Physics

Posted by Phiphy on 09/20/2009

Time: 9/19/2009

Speaker: Chris Brust

Outline:

  • Definition of groups: 4 axioms.
  • Finite groups: defined by a multiplication tables.
    Example: Permutation group S_3
  • Group representations: mapping a group to a set of matrices.
    trivial rep, faithful rep, reducible and irreducible rep, unitary rep
    Example: 3 (and the only 3) irreducible rep of S_3, 1 faithful but reducible rep of S_3
  • Lie groups: continuous groups which can also be described as a manifold.
    Commonly used Lie groups in physics:
    U(N), SU(N), O(N), SO(N), L(N), GL(N), Sp(N), E_N
  • Group properties: Isomorphism(btwn two groups), Abelian and non-Abelian, compactness, connectedness, simply connected or not.
    Examples: U(1), SU(2), O(1,3), Poincare group
  • Lie algebras: defined by 3 axioms.
  • Generating a group from an algebra and vice versa.
    Example: Heisenberg algebra and group

Comments:

  • Very nice talk, informative and well organized. Thank you, Chris.
  • Why do we need group theory in physics?

It’s all about symmetry. Symmetry plays a significant role in modern physics. From crystal lattice in condense matter to elementary particles in high energy physics,  it is its symmetric structure that causes the system’s rich phenomena, and almost all we care about in theory is related to its symmetry realization and breaking. Group theory is an indispensable math tool for describing symmetries. A system has some symmetry means the Hamiltonian or Lagrangian is invariant under the transformation of the corresponding group. So we get some rigid mathematical form for this symmetry and studying this system becomes studying the Hamiltonian or Lagrangian under such constraints. Buy using this tool, we can even lift different specific physical systems to some abstract structure and find their common properties, as happened again and again in the history of physics. One of the most remarkable example is the 2008 Nobel price for physics: Nambu was awarded for his work on spontaneous symmetry breaking in superconductor which latter played a vital role in particle physics.

  • Why are group representations so important in physics?

Groups are only some abstract math f0rms. To connect math to physics, we need one more step: to find some specific representations of the group. Different systems may have the same symmetry, but their constituents can have very different behaviors under the symmetric transformation. Some may not change, some may exchange identities with each other, some may shift by some values, but the Hamiltonian or Lagrangian is invariant under all these changes. In math language it means, they are in different representations of the same group. For example, the ones that are kept unchanged are in trivial representations, ie., I. In particle physics, the role of representations is even more obvious: the nature has only one fundamental physics law, which means the groups that describe all the matters in the universe are the same, but why are there so many different species of fundamental particles with different spins and interactions? They are distinguished by different representations. Different spins and momenta are distinguished by rep’s of Poincare group, different interactions are distinguished by rep’s of gauge groups.

  • Why do we also need Lie algebra?

There is a most important class of groups called Lie groups, which played a central role in studying QFT. Lie groups describe continuous symmetric transformations, eg., Lorentz transformation, translation and gauge transformations. However, usually we only care about *local* properties of a system, ie., how it behaves under some infinitesimal transformations. That’s where Lie algebra comes out. In geometric language, Lie groups can be taken as manifolds, each group element is a kind of ‘translation’ on the manifold and the generators of a Lie algebra are a set of basis of the manifold. (To imagine it, you can use ordinary vector space as a analogy.) By studying the properties of the basis, we can know the properties of the whole manifold, but wait, not all properties of the manifold are included in the basis. The same Lie algebra may generate different Lie groups. For example, SO(2) and U(1) are equivalent both as Lie algebras and groups (ie., they are isomorphic); While SO(3) and SU(2) have the same algebra, they are different groups (SU(2) is simply connected but SO(3) is not. ). This is because some discontinuous symmetry distinguishes their global properties[1]. Fortunately, in QFT, usually the local property says everything about physical observables we care about, eg., cross-section of collision, life time, etc. So we do not need to be too serious on distinguishing Lie groups and Lie algebras.

  • The first step of constructing a quantum field theory
    - One example of group representation theory used in QFT

One of the most important Lie groups in QFT is of course the Poincare group, which carries the physical meaning of special relativity. To make a relativistic quantum mechanics, we only need to let each of the group elements act on a vector(state) of a Hilbert space which satisfies all the axioms of quantum mechanics and get another vector in the same space,

{\Psi }' = e^{-ix_{\mu}P_{\mu}-i\omega_{\mu \nu} J^{\mu \nu} }{\Psi}

That means this Hilbert space is a symmetric space under the transformations of the group. So we have relativity and quantum mechanics both satisfied. Then our task is to find all the possible representations of the group and do experiments to see what representations are chosen by the nature, ie., what species of particles do exist in nature. Mathematically we can prove that translation and Lorentz transformation can be disentangled, and further, the only irreducible representations of Lorentz group are spin-half-integer particles. Now we find all possible kinds of elementary particles in the nature! (Assuming relativity and quantum mechanics are correct, of course.) In reality, we see only spin-1/2 , spin-1 and spin-2 elementary particles, but who knows spin-0 and spin-3/2 elementary particles exist or not, they may be waiting for us on the LHC[2].

Till now we only discussed Poincare symmetry for free particles. Most of the interactions are related to gauge symmetry and they can be studied in a similar manner.

Now we’ve learned the first step of constructing a general quantum field theory: determine all the symmetries of the system, find and select certain representations of the symmetry groups, and write down a Lagrangian which is invariance under the symmetric transformations by using these representations as degrees of freedom.

—————————————————

[1] For your information, SO(3) is in fact isomorphic to SU(2)/Z_2, where Z_2 means an action of orbifold. )

[2] Elementary particles with spins higher than 2 are theoretically forbidden for some deeper reasons.

[3] Text books on group theory suggested by the speaker:

Georgi, Lie Algebras in Particle Physics

M.S. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory: Application to the Physics of Condensed Matter

Micheal Tinkham, Group Theory and Quantum Mechanics

Posted in Communications | Tagged: , | 1 Comment »

Nima’s Talk

Posted by Phiphy on 12/13/2007

Nima Arkani-Hamed may be the most famous and interesting name in the new generation of physicists. I attended his talk today in UMD, and his words are so impressive that even though I can’t understand the details, I feel deeply interested and attracted.

His title is ‘quantum gravity: possible and impossible’. In all, his purpose is to convince us that because of gravity, IR and UV are not totally decoupled as we thought before, gravity can contribute to high dimensional operators in EFT, and that not all IR effective field theory have a consistent quantum gravity theory as its UV completion, no matter it is string theory or not. The most important criteria he uses is the Weak Gravity Conjecture, which says that gravity must be the weakest long range (U(1)) force, and an effective theory should break down even below plank scale, at gM_{pl}, where g is the gauge coupling, otherwise this system will decay into a black hole. A sharper form of WGC is that any matter must have M/Q<1. So if an IR theory violate WGC, it can never find a consistent quantum gravity theory as its mother theory.

So what EFT may be on the ‘dead’ list? One most important example is the inflation models which produce gravity wave, such as chaotic inflation. If gravity wave is generated, we require that the scalar field vev must be greater than plank scale. Naively, this can be realized in a 5 dimensional model with a very small 5D gauge coupling and it seems very natural. However, no one can find it in string theory, no matter what form they tried, there is always something, such as diliton, coming out to destroy it. Is there any general reason why they failed? The answer may be yes: it violates week gravity conjecture. There is another model named ‘N-flation’, with N species of scalar field triggering inflation. It is also problematic because it violates species bound which comes from constrain of entropy of a black hole. Moreover, if considering loop correlation of G_N, we have \frac{1}{G_N} \sim \frac{N}{l^2} which also leads to violation of WGC for large N. So Nima boldly guess that we can not see gravity wave in CMB, otherwise, it will be a big crisis.

Another interesting model he talked about was Euclidean warm hole. It was proved by Coleman that warm hole theory can be a local theory with global symmetry breaking, which can be realized with axion coupling with gravity. However, as chaotic inflation, we can not find it in string theory. And the obstacle is also WGC. To realize a warm hole, we just need to find a geodesic longer than 1/M_{pl}, which means we have a energy scale higher than plank scale and the corresponding U(1) gauge coupling is weaker than gravity. Further, he said that if we realize it in a AdS/CFT world, it will violate unitarity of CFT, which I do not understand.

Nima also talked about some other models, checking them with WGC. Especially, he mentioned RS. He said we can not ignore the S^5 compacted in AdS_5 when we go down from string theory. Otherwise if we take a brane near the UV side with a string on it, the corresponding particle(gauge boson) will have mass M>1/L_{AdS} which can be larger than M_{pl}. We need the extra-extra dimensions to ‘dilute’ the mass of the particle. I do not understand how either, too ’string’.

Nima is really a good speaker. He makes each point very clear, and goes through the details very smoothly. Speaks fast, writes faster.

Posted in Communications | Tagged: | Leave a Comment »

Also Sprach David Gross

Posted by Phiphy on 12/06/2007

David Gross was invited to our department for 3 days. During one public lecture, one colloquium and some informal chat, there are something he said which may be interesting to hear. One thing to remind you, Gross is a very conservative reductionist, so you may not like some of his words. Welcome to leave your comments. I just quote them (not word by word) in a random order:

Posted in Communications | Tagged: | Leave a Comment »

Things I learned these days

Posted by Phiphy on 12/03/2007

  • Neother Theorem is the Lagrangian form of Ehrenfest Theorem which is in Hamiltonian form. There are two correspondences (not very exact) :
  1. The Lagrangian is invariant under some symmetry corresponds to the Hamiltonian commutes with some operator (symmetry group’s generator)
  2. The Neother charge is a time constant corresponds to the expectation value of the symmetry operator is a time constant, actually, the Neother charge is just the expectation value of the generator.
  • From Kaplan’s Colloquium (11.29):
  1. EW theory must break down because of 4W interaction, just like 4 fermion interaction must break down, which is caused by violating unitarity. And the solution is similar, 4 fermion need a gauge boson to change 4 vertex to 3 vertex, and 4W need a Higgs (or other new particle?) to change 4 vertex to 3 vertex.
  2. MSSM has more than 100 parameters! ( I heard this for the first time)
  3. Higgs mechnism can be traced back to Schwinger, who proved that massive gauge bosons do not necessarily violate gauge symmetry, by introducing a scalar field, but did not mention symmetry breaking. And there were other guys’ work following this, showing symmetry breaking, until Higgs pointed out the existence of a scalar particle following the referee(highly suspected to be Schwinger)’s suggestion. So came the name ‘Higgs particle’.
  • From Tom’s seminar (12.3)

SUSY can be broken explicitly at UV (elementary sector) but emerges accidentally at IR (composite sector). The symmetries at lower energy are more than that at higher energy, which seems blizzard, but not. From 5D view, IR have more gauge symmetry and so more degrees of freedom just because UV and IR are two vacuums separated by the bulk (or domain wall); from 4D view, the different symmetries become global symmetry so it will not take more degrees of freedom, and the low energy global symmetry is always broken at high energy. (Because of gravity)

Posted in Communications, Sparks of the mind | Tagged: | Leave a Comment »

大牛现身

Posted by Phiphy on 08/01/2007

今天一打开arXiv，就发现 ‘t Hooft 同学挂了两篇帖子。我还以为是他新发paper了，原来都是报告发言稿。其中一篇The Grand View of Physics，短短6页，从Salam的高度和胆量谈开去，为的是再次阐明自己对以量子力学为基础的从量子场论到超弦的一系列理论的怀疑。他说建立在Hilbert空间线性变换上的量子力学很可能是不完备的，我们仍然需要努力寻找量子力学“背后”（尽管很多人认为量子力学没有“背后”）的解释以发现可能的全新物理。不知道他是否真的接爱因斯坦的班，是个彻底的决定论者。不过退回到量子力学重新寻找全新的道路是个极其艰巨的任务，恐怕也只有’t Hooft 这种功成名就的老前辈才敢玩。无论如何，’t Hooft这种牛人存在的价值就是让人大开眼界。还有一篇比较长，Emergent Quantum Mechanics and Emergent Symmetries. 只看了abstract, 把量子统计涌现性（根源是决定论的）跟规范对称性和广义坐标变换等价性联系起来，很难说有多大理论价值，但貌似很有意思，什么时候有空再看看。

Posted in Read them | Tagged: | 2 Comments »

Web resources for extra dimensions(XD)

Posted by Phiphy on 07/30/2007

A brief review of XD: Backreaction

An introduction to RS Model: Flip potato

Lecture Videos on XD:

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Higgs Cartoon

Posted by Phiphy on 04/06/2007

A very intuitive portray of Higgs by comparing with condensed matter physics. Interesting, isn’t it?
From http://www.hep.ucl.ac.uk/~djm/higgsa.html

A quasi-political Explanation of the Higgs Boson;
for Mr Waldegrave, UK Science Minister 1993.

1. The Higgs Mechanism

Imagine a cocktail party of political party workers who are uniformly distributed across the floor, all talking to their nearest neighbours. The ex-Prime- Minister enters and crosses the room. All of the workers in her neighbourhood are strongly attracted to her and cluster round her. As she moves she attracts the people she comes close to, while the ones she has left return to their even spacing. Because of the knot of people always clustered around her she acquires a greater mass than normal, that is, she has more momentum for the same speed of movement across the room. Once moving she is harder to stop, and once stopped she is harder to get moving again because the clustering process has to be restarted. In three dimensions, and with the complications of relativity, this is the Higgs mechanism. In order to give particles mass, a background field is invented which becomes locally distorted whenever a particle moves through it. The distortion – the clustering of the field around the particle – generates the particle’s mass. The idea comes directly from the Physics of Solids. Instead of a field spread throughout all space a solid contains a lattice of positively charged crystal atoms. When an electron moves through the lattice the atoms are attracted to it, causing the electron’s effective mass to be as much as 40 times bigger than the mass of a free electron. The postulated Higgs field in the vacuum is a sort of hypothetical lattice which fills our Universe. We need it because otherwise we cannot explain why the Z and W particles which carry the Weak Interactions are so heavy while the photon which carries Electromagnetic forces is massless.

The Higgs vacuum
(Distributed in the universe like aether, but must be covariant – scalar)
Particles coupled with Higgs gain their masses
(The stronger the coupling is, the more massive the particle is)

2. The Higgs Boson.

Now consider a rumour passing through our room full of uniformly spread political workers. Those near the door hear of it first and cluster together to get the details, then they turn and move closer to their next neighbours who want to know about it too. A wave of clustering passes through the room. It may spread out to all the corners, or it may form a compact bunch which carries the news along a line of workers from the door to some dignitary at the other side of the room. Since the information is carried by clusters of people, and since it was clustering which gave extra mass to the ex-Prime Minister, then the rumour-carrying clusters also have mass. The Higgs boson is predicted to be just such a clustering in the Higgs field. We will find it much easier to believe that the field exists, and that the mechanism for giving other particles mass is true, if we actually see the Higgs particle itself. Again, there are analogies in the Physics of Solids. A crystal lattice can carry waves of clustering without needing an electron to move and attract the atoms. These waves can behave as if they are particles. They are called phonons, and they too are bosons. There could be a Higgs mechanism, and a Higgs field throughout our Universe, without there being a Higgs boson. The next generation of colliders will sort this out.

A fluctuation in Higgs vacuum


Condensation of Higgs Boson

from David J. Miller, Physics and Astronomy, University College London.
(cartoons courtesy of CERN).

Posted in Physics is fun | Tagged: | Leave a Comment »

Why off-shell?

Posted by Phiphy on 04/04/2007

Why are the virtual particles off-shell? One common explanation is uncertainty of energy. My opinion is just the opposite – it’s because of the certainty of energy, for the energy-momentum must be conserved. Look at the vertex in the Feynman Diagram, if we determine the four momentum of the incoming particle and the outgoing particle, the four momentum of the exchanged virtual particle is fully determined by conservation of energy-momentum. Compare with the 2-body decay, where the final momentum magnitude and energy are exclusively determined by conservation law and on-shell condition, in the virtual particle case, the final states are arbitrary, so for most of the cases, it is impossible for the virtual particle to be on-shell.

For this intuition, the virtual particles are really unreal. I would rather interpret them as transforming of quantum numbers than particles carrying these quantum numbers? What’s the difference? Maybe not too much. Virtual particle is at least a convenient concept for calculation though it seems wired.

Update:
So what happens if we set this propagating ‘particle’ to be on-shell? By using conservation of four momentum and on-shell condition, the scattering angle is entirely determined. The result comes out to be: for Coulomb scattering, the angle is 0, for which the differential cross section blows out; for s channel process like ee->mumu or Compton scattering, the on-shell condition can never be reached. Then we see the xx-shell has nothing to do with virtual ‘particles’, and the blowing out is expected because on-shell condition is a pole for the propagator.

Further more, on-shell condition is equivalent to equation of motion for free particles. When we calculate propagator, we put a delta function on the other side of the equation, so it’s no longer free. I should have noticed this long before! off-shell is equivalent to interaction. All interacting particles are off-shell. Why outer-leg particles are on-shell? Because they are far away from each other and are taken as free. This fundamental reason leads to the direct reason for off-shell which I discussed above: transformation of momentum and energy.

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