My book on physics... (the second chapter)

Discussion in 'Free Thoughts' started by Reiku, Dec 28, 2011.

  1. AlphaNumeric Fully ionized Registered Senior Member

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    6,699
    Except that if you don't understand the example you're getting from somewhere you don't know if you haven't introduced or copied an error. For example, for ages you would say that the normalisation condition for a wave function is \(|\psi|^{2} = 1\) when anyone who knows what the inner product on the wave function is would know that to be wrong.

    And this is ignoring the fact the introduction of such concepts to a 16 year old is inappropriate. They don't know index notation so using it is pointless. They don't know matrices so referring to them is pointless. If you really wanted to introduce high level concepts then you have to them in such a way as to not require any knowledge a 16 year old wouldn't have already been taught. This is why pop science books use analogies, rather than the proper formula.

    Saying \(\mathcal{L} = \partial_{\mu}\phi \partial^{\mu}\phi^{\dag} + \bar{\psi}\gamma^{\nu}\Big(i\partial_{\nu} - A_{\nu})\psi + \bar{\psi}|\phi|^{2}\psi - \frac{1}{e^{2}}F_{\rho\sigma}F^{\rho\sigma}\) to a field theorist is enough to tell them all about the model you're talking about. Say it to a 16 year old and its meaningless.

    Or how about spin? Why is \([S_{i},S_{j}]=\epsilon_{ijk}S_{k}\) viable but \([S_{i},S_{j}]=\epsilon_{ijk}S_{k}-\delta_{ij}S_{1}\) is not? Just stating the first case is valid doesn't tell you why its valid, you aren't providing insight into what you're talking about. This isn't a rhetorical question, I'd like you to answer.

    If you were trying to write a pop science overview I'd not have a problem but you're trying to pepper it with inappropriate, both in terms of content and audience, mathematics.

    Except you won't be providing any real benefit over reading a pop science book because none of the mathematics is given sufficient explanation.

    I remember the end of my 1st year where I looked at the 2nd year homework questions for the basic QM course and thinking "How does any of this notation work?! How does anyone understand it?". I tried to put the various symbols in contexts I'd already covered but it still didn't make sense. Looking back now I can see just how far off the mark my attempts at understanding were but it wasn't until I did the details properly I saw how wrong I'd been. And I'd got a year of university mathematics under my belt, including vector calculus and linear algebra!

    You cannot pass a university course by reading pop science. And by 'you' I mean anyone, including you. Even a few equations here and there won't help. One of the books I read when I was 17 included Schrodinger's equation, Planck's formula, the Uncertain Principle equation, various equations due to Einstein, the Hydrogen spectra, I could write them all down from memory and talk about them easily but I didn't understand them because I hadn't had the notation explained to me or the mathematical foundations upon which they were built. Aside from the "Oh, I've heard of that!" moments during the QM lectures all that pop science + seeing all the relevant equations counted for nothing when it came to actually working with the equations.

    I say this because I am reiterating two points. Firstly that you aren't in a position to walk the first year of a physics degree. You haven't even demonstrated you could get onto such a course. And secondly that throwing in equations to a wordy description doesn't necessary add enlightenment. I get it, I understand how you can have the impression of understanding it, I was naive like that once too. It got smashed out of me at university when I realised just how completely different actually doing the details on a working level is. I'm sure the other science graduates here will agree. Pop science gets you interested enough to work through the boring details till you get to the good stuff, it doesn't actually provide you with enough information to start there.

    If you ever get yourself onto a physics degree you'll realise this. But I am wondering if your lack of reply to my questions about when, if ever, you plan to get off your arse and start working towards that physics career you've mentioned means that you don't ever plan to do that. Are you going to try to get onto a physics degree course at some point or will you still be doing this when you're 35?
     
    Last edited: Dec 31, 2011
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  3. Reiku Banned Banned

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    Chapter Three
    Quantum Mechanics, the Bigger Picture


    So, up until now, we have been able to realize that the world at large, indeed the universe at large is built of large galactic structures, supergalaxies that are made up of smaller galaxies which are in constant attraction with each other through gravity is in fact made up of much smaller world of atoms and subatomic particles. Now we will investigate the fields which govern these particles, the world of quantum mechanics and it's very interwoven brother, quantum field theory which attempts to unify it all.

    We won't actually talk about any unification attempts yet however, simply because there is a whole chapter dedicated to that cause but we will make a short mention of one later in this chapter.

    Quantum Mechanics was the first scientific transition from classical physical theories to the modern understanding. Quantum Mechanics readily allowed us to understand that the world at large, (the world of macroscopic interactions) was vastly different to the quantum world. In fact, it is often said that there is a cut-off where the quantum world ceases to apply to the world of the very large.

    One quantum weirdness that does not apply (or atleast cannot be seen) at the world of the large is the wave function (given by this mathematical symbol \(\psi\))which seems to be a physical manifestation/phenomenon. [1] [2] You might remember we covered a little about the wave function. It is a probability field, or as some like to call it, a P-Field. This probability field manifests the likelihood of finding an object anywhere in spacetime.

    Now, it is not that the wave function does not apply to macroscopic objects however. The wave function actually does, but it's wavelength is incredibly small for objects that are much larger than the quantum world. The physical wave function was actually observed by scientists before the first citation provided only moments ago. The reason why we even began speculating about a wave function was because of quantum mechanics and because of what is called the dual nature of matter.
    The dual nature of matter does what it says on the tin. A particle actually has two physical natures: Through careful experimentation, we were able to deduct that a particle has both a wave and a particle nature. Sometimes, a particle would act like a wave as it moved through spacetime, but as soon as something came along and observed the object, the wave function would collapse and what would be measured would be a tiny pointlike particle.

    The first time this was measured was by a physicist called Thomas Young in what is famously known today as the Double Slit Experiment. In this crafty little experiment, the scientist would shoot photons from a gun until only a few photons where being shot at a time. The photons where being shot at a screen which had two slits - an upper slit and a lower slit. Either the lower slit or the upper slit can be shut at any point in time.

    Now normal logic would suggest (and this is where quantum weirdness is really understood) that if both slits where open then one might think that more particles would reach the screen behind it... but to the surprise of physicists more particles would reach the screen (at the end of their journeys) as tiny little dots when one of the slits had been closed than there would have been if both slits had remained opened.

    Counterintuitive? Indeed it is!

    It must be said, it baffled physicists for some time until an answer came about. But the answer was quite weird. It seemed that to explain this very strange phenomenon required that matter did not act like individual particles cascading and scattering in space. It required that they rather acted like waves some of the time, where the waves could interfere with each other, much like waves in the water as they undulate and effect each other when they come into contact with each other. So as two particles might be heading towards the two slits, they would be spread out over some space until they reached a narrow width in which they would interfere with one another.

    These waves where called the wave function - a smear of probabilities over some region of space and time. Enter quantum mechanics, a completely new description of particles which had both the wave like properties, but still had particles properties as well... afterall, when they reached the screen, they were still pointlike dots. We soon realized, quantum mechanics was not bound to the strange wavelike nature alone. There was in fact many more strange features that would come into the working theory. One such feature was a phenomenon called quantum tunnelling.

    This feature was a consequence of the wave particle duality of particle nature and is explained through a cornerstone of physics we briefly covered in our last chapter, the Heisenberg Uncertainty Principle.

    The idea was devised from studies of radioactivity. It was discovered by Henri Becquerel in 1896 and it in very short terms, is a way for a particle to tunnel spacetime when a barrier is encountered. It is a bit like myself, if I decided to walk to a nearby mountain (that is if you live in any mountanous area's or even where there are some steep hillsides) and instead of walking around the object, I could simply tunnel my way through. Easy way of travel no? It would save me a lot of energy?

    Well, this is exactly the same reason a quantum particle might do it. There is a law in nature called action (we will cover this more in The Dirac Equation part) and the action of an object is given by some particle moving from one place to another in terms of it's kinetic energy. But what particles will do is execute actions which will require the least amount of energy!

    This condition of nature is called the Least Action Principle. It will cost less for a particle for instance, to simply move through a barrier of spacetime (those quantum hills and mountains) than it would to simply try and move around it.

    It turns out that there are two quantum states which help define a system which is using the least amount of energy. The system which corresponds to the least amount of energy is a particle which is in its ground state. The ground state particle is usually a very stable particle (meaning it will not radiate superfluous amounts of energy and will not prone to decaying quickly into new types of particles). A particle which is ripe to radiate away energy and jump to higher levels of energy states is called an excited energy state. A good example, one used by Bohr (the Father of quantum Mechanics) is said to have used room temperature conditions as a good example of a room made from ground state atoms.

    Ground State atoms simply rearrange their particle constituents (the electrons) in such a way that the energy of the atom can be translated as being a ground state. The ground state electrons remain as tightly bound to the nucleus as they can (therefore they exist in the orbits closest to the atomic nucleus in a wave function of states). When one of these atoms are ever excited, they undergo a quantum transition of energy states called a ''quantum jump.''

    An interesting part about observer physics, is that a particle ripe to radiate away it's energy can in fact be suspended in time, infinitely stable if one wished. If someone came along and observed an atom ready to give up it's energy observed the particle periodically over some time, the observer would effect the quantum evolution of the object; in fact, it means in a more technical sense that the zeno effect is the surpression of unitary time evolution itself. In fact, so apart of the world the zeno-effect seems to be, is that birds have themselves magnetoreceptors which acts as a birds magnetic compass.

    The Universe could be in an excited state. Again, along the same principles if our universe is in an excited state, then it should not conserve energy very well. If our universe is indeed expanding and the recession of the most distant of galaxies relative to us in the observable universe, then acceleration requires more energy and this would mean the universe is slowely using it's own up in some form of an excited state. There is every possibility though that the universe began in a ground state and has evolved and become excited over a certain length of time.
    There are many more quantum mechanical concepts we could have covered but I now want to move onto quantum field theory itself. It is much more interesting anyway, it must be said.

    Quantum field theory was developed during the 1920's and many notable scientists of that decade where involved in assembling the framework of modern quantum field theory. Some of these scientists were Werner Heisenberg, famous for his Uncertainty Principle, Max Born who was a physicist who is famous for developing his probability density postulate \(\int |\psi|^2\) (that the square of the absolute wave function \(|\psi|^2\) is the probability of finding an object in a certain quantum state) which was made of good use in the Copenhagen Interpretation of quantum mechanics [3]. There was also Pascual Jorden who formulated the ability to canonically quantize Harmonic Oscillators (harmonic oscillators are just particles that are the quantized field theory) [4].

    There were many more contributors to quantum field theory, including Albert Einstein, probably one of the most famous scientists who had ever lived. It was his theories on relativity which helped to pave an understanding of electromagnetism in the context of quantum fields.

    Paul Dirac made a huge contribution to quantum field theory - he created an equation which could satisfy quantum fermion fields: Particles which obey spin 1/2 dynamics and the best part was that it was completely relativistic so it was a way to unify quantum field theory with the relativistic theory and satisfy what is called Lorentz Invariance. [5] We will cover his equation at some point in this chapter. His equation has much to say and is a wonderful peice of mathematics.

    Quantum Mechanical models are parametrized by usually an infinite number of degrees of freedom. In the language of quantum field theory, the freedom is made up of quantum fields which permeate all of spacetime. The quantum fields are the fields in which distortions take place, namely particles. Particles are then quantum field fluctuations.

    The fluctuations in field theory states that is it is effective self-energy of a field. The self-energy is denoted as \(\sum\) and represents the contributed energy of a particle which basically means that it is energy gained from interacting with it's own quantum field. The particles which make up such a field, is called the fields quanta. Every field had to be quantized, no less. Doing so we find that the quantization of an electromagnetic field is the photon for instance.
    Despite the remarkable success of quantum field theory, there was however a few problems along the way, such as the self-energy of an electron. The excited energy states of an electron in the presence of it's own field yielded an infinite energy. Some believe that the method which attempts to solve this problem today is probably not the correct way to tackle the problem. The solution that mathematicians used was a technique called ''Renormalization.'' Using other infinities (in a very loose and incorrect way) is a way for us to subtract them from other infinities in order to produce finite quantities.

    As it turns out however, there are many infinities in physics and are considered by most as a physical breakdown in the theory. The one who expressed his idea's on renormalization was Paul A. M. Dirac, the same scientist we briefly covered who created the Dirac Equation which could describe spin 1/2 fermion fields. One of Dirac's most famous rebuttals against the postulates of renormalization was:

    ''Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small - not neglecting it just because it is infinitely great and you do not want it!''

    And it made sense. Why should we throw away an infinity just because it is very large? It turned out that renormalization would be quietly accepted over the remaining years to present date. The progress was made after realizing that all infinities in quantum electrodynamics are related to two main effects, the self-energy of the particle [6] or the vacuum polarization. Interestingly it was Diracs own equation which helped understand how vacuum polarization would work in nature, but first, what is vacuum polarization?

    It is the realization that the vacuum is in the most simplest definition, a world were virtual particles roam free all the time. In the language of the electromagnetic physics, it is the background EM-field in which electron-positron pairs of particles are created and interact with their own fields. Because of this background electomagnetic energy (called virtual particles), we can no longer have what is called empty space. Indeed, all space and time is filled with quantum energy in the form of long lived or short lived phenomena, the latter being provided from these phantom particles we called virtual for obvious reasons.

    Dirac, in the formulation of his field theory on fermions, like electrons, required some kind of renormalization. In order for an electron not to simply fall back into the vacuum, there needed to be a virtual particle in the vacuum which would prevent it from falling an energy level because it would be occupied by a quantum state. It turned out his theory would have required an infinite positron sea of particles, (that is an anti-electron). These negative particle's would make a huge sea and a huge problem for Dirac. An infinite charge filling quantum field permeating all of spacetime was not a good thing. The problem was solved by renormalization, of course, the theory which Dirac did not agree on all too well.

    The bare vacuum would have needed to have an infinite positive charge as well which would exactly cancel out negative charge. In the end, it was decided that a reformulation rule applied by quantum field theory was necessary to understand the Dirac Sea by treating the positron as a particle which was ''not the absence of a particle''. The reason why this is important is because Dirac interpretated the positron as a ''hole'' in the vacuum.It must be noted however that an important feature of the reinterpretation rule is that it does not eliminate the infinite energy issue of the vacuum. Only some renormalization technique could sort this out still.
    So as one can surmise, quantum field development has been a tricky feat. That does not go to say however we have a complete working theory. There are many parts of quantum field theory which still make little sense to us. Attempts at unification began with Einstein after he developed his General Relativity Theory, but he never suceeded, mostly due to the fact that he had an incomplete framework, namely the strong force was not known at the time.

    Relativity was big at the time, at the core of the quantum mechanics and the developing field theories. Dirac was no doubt one of the largest contributors to the quantum field theory development and we shall now cover his most famous equation of all time, which would help describe particles called fermions to almost near accuracy.

    The Dirac Equation

    The Dirac Equation is a relativitistic wave equation which is analogous to the Schrodinger Equation (the standard wave equation in physics). The Dirac Equation attempts to make a prediction called antimatter. At first, when Dirac came across the solution of a particle having an antiparticle, he never admitted to it's existence. Afterwards when asked why he never took it seriously, he is qouted as saying that he simply was not ''brave enough''.

    It predicted an antiparticle that was the ''mirror'' identity of the normal particle but with a change in sign of the charge (in respect to electrons, which have a negative charge) positrons possess a positive charge. So one must expect a sign convention change when speculating the existence of the two possible states, (the particle state) and (the antiparticle state).

    In Relativity, the way to express the relationship between momentum and the energy of a system is

    \(E^2 = p^2c^2 + M^2c^4\)

    Turning this into natural units [7] we would have

    \(E^2 = p^2 + M^2\)

    It will be conventional later to also assume that the energy will be replaced by the quantum field energy operator \(i\hbar \frac{\partial}{\partial t}\) and the similar for the momentum for it's respective operator \(-i \hbar \nabla\).
    We shall keep this in mind - now, right moving particles and left moving particles are basically particles and antiparticles. We are not sure why we have left moving and right moving particles other than it is a fact of nature. Right movers in Dirac's theory is given as (still using natural units)

    \(\frac{\partial \psi_R}{\partial t} = -\frac{\partial \psi_R}{\partial t}\)

    Left moving waves are given by

    \(\frac{\partial \psi_L}{\partial t} = +\frac{\partial \psi_L}{\partial t}\)

    The coupled form of these equations comes in the form

    \(\frac{\partial^2 \psi}{\partial t^2}= \frac{\partial^2 \psi}{\partial x^2}\)

    A neat mathematical way to express two waves can be given as

    \(\displaystyle \binom{\psi_R}{\psi_L} = \Psi\)

    Which is a column vector.

    Now these two equations might look strange to you, like notation using

    \(+\frac{\partial \psi_L}{\partial t}\) but don't worry, it can be understood if you take into consideration that the partial sign \(\partial\) is informing you of a ''change in'' \(\psi_R\) or \(\psi_L\) with respect to a change in either position \(x\) or time \(t\):

    we begin with a function

    \(\psi = e^{i(kx - \omega t)}\)

    \(\omega\) here is the angular frequency and \(k\) is the wave number, two important observables of the equations. The angular frequency and the wave number is related by a number called the ''phase velocity'' and comes in the form of
    \(\frac{\omega}{k} = V_p\)

    Suppose we return to a new form of our wave equation

    \(\frac{\partial \psi_R}{\partial x} + \frac{\partial \psi_R}{\partial t} = 0\)
    If we pull down a the derivative in respect to \(x\) in our function \(\psi = e^{i(kx - \omega t)}\) then we pull down a \(ik\). If we pull down the derivative with respect to \(t\) then we obtain \(i \omega\). This soon led to the notion that \(i \omega - ik = 0\).

    In Einstein's theory, he developed momentum in terms of \(p\). In Dirac's theory, \(k\) is the momentum and \(\omega\) has dimensions of energy \(E\). The two corresponding equations are

    \(\omega^2 = M^2 + k^2\)

    \(E^2 = M^2 + p^2\)

    You can clearly see how they interchange one anothers meanings. Taking the time derivatives for fields will result in a notation using an upper case dot \(\dot{\psi}\). This simply reads as \(\frac{\partial \psi}{\partial t}\). Dirac under his theory, required the use of matrices and why will be explained soon. But for now, assume a newcomer called \(\alpha\)

    \(\dot{\psi} = \alpha \frac{\partial \psi}{\partial x}\)

    But what we want to is to rewrite this equation in a more compact form so what we will do is realize some of the things which is happening in here. When you take a quantum wave like \(\psi\) and hit it with \(\frac{\partial}{\partial t}\) it might be seen from the function we had earlier that it pulls down an \(i \omega\). This would mean that the \(\frac{\partial \psi}{\partial x}\) part pulls down an \(ik\), so we see that we quickly retrieve our equation again \(-\omega^2 = -\alpha - k^2\) but with the added matrix and without any mass term. Cancelling the minus signs and adding a mass into our system is now beginning to look more like the Dirac Equation

    \(\omega = \alpha k + M\)

    A new matrix is required however and again, but for now we will just accept it and it get's attached to the mass term \(M \beta\). The justification for a new beta matrix and the previous alpha matrix comes from a strange fact concerning the algebra of Dirac Equation.

    Known as Clifford Algebra's, they concentrate on the anticommutation relations that occur in the theory. We covered a few basic examples of commutator relations in the previous chapter when talking about commutation relations between certain observables, such as position \(x\) and momentum \(p\). The first requirement of the matrices is that they satisfy \(\omega^2\) so we will have

    \(\omega^2 = \alpha^2k^2 + M^2\beta^2\)

    so we have had to square the matrices along the way you might have noticed. An algebra method of decomposing this equation can make

    \(\omega^2 = (\alpha k + M \beta)(\alpha k + M \beta)\)

    Now we are going to expand the right hand side of this equation

    \((\alpha k + M \beta)(\alpha k + M \beta) = \alpha^2k^2 + M^2 \beta^2 + \alpha \beta k M + \beta \alpha kM\)

    Since we know the matrices require that \(k^2 + M^2\) then we can see the first term \(\alpha^2k^2 +M^2 \beta^2 \) already satisfies the equation, so the rest can be neglected \(\alpha \beta k M + \beta \alpha k M\). Because of this however, we can clearly see that there is some kind of commutation relation happening in the equation, because \(\beta^2=1\) would need to be satisfied and also \(\alpha^2 = 1\) would need to be satisfied, again because \((\alpha \beta + \beta \alpha)km\) becomes invalid.

    The alpha matrix is given as

    \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \alpha\)

    so \(\alpha^2\) is given as

    \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

    \(\beta\) is not the same as this because it satisfies the clifford algebra \(\alpha \beta + \beta \alpha = 0\). This means it's matrix has the form of

    \(\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \beta\)

    The equations Dirac arrived at to express right moving and left moving waves were

    \(i \dot{\psi} = -i \partial_x \psi_R + M\beta \psi_R\)

    ... are right moving waves. I have already explained the notation for that strange looking dot over the psi wave, but I haven't explained the notation involving \(\partial_x \). This is only a simplification notation for a partial derivative, in this case as \(\frac{\partial}{\partial x}\). The left moving waves are given as

    \(i \dot{\psi} = +i \partial_x \psi_L + M \beta \psi_L\)

    The mass term allows us to couple these equations and make what is known as a Majorana Equation, which describes particles as their own antiparticles.

    \(i \dot{\psi} = (-i \alpha \partial_x + M \beta )\psi\)

    And this is the Dirac Equation. As I said before, it predicts positive and negative state's of matter. It also accomodates for quantum spin, that pesky subject we covered in the last chapter involving Pauli Spin Matrices. The Pauli Spin Matrices also enter into the compartments of the \(\alpha\) matrix, for instance

    \(\begin{pmatrix} \sigma_j & 0 \\ 0 & \sigma_j \end{pmatrix} = \alpha\)

    Each entry is a 2x2 matrix and \(\sigma_j\) is the Pauli Matrix where \(j=(1,2,3)\). Now, his equation can help manifest a better understanding of the dynamical nature of systems at relativistic energies as well as rest, of course. In relativistic terms, relativistic energies are concerned with particles which possess a large kinetic energy term. For particles which are in their ground states, they will possess a low kinetic energy and will tend to be more at rest. We will cover ''rest'' in more detail in the Relativity Chapter.

    Speaking of kinetic energy, when you take this quantity away from the potential energy you get what is called the Kinetic Potential. This is just a fancy name for what is more practically named, The Langrangian of the system. In a very simple way to put it, just for the sake of teaching some of it's simple textbook structure is:

    \(L = T - V\)

    Explaining it in terms of quantum fields however is usually quite a difficult task and would require extensive research into the area, however for the benefit of the reader we can cover a Langrangian which will not be too hard to understand because it will be primarily the Dirac Equation but in a new guise. This is just my own attempt at trying to make the information I have gathered over the years which try and explain these dynamics in the most simplest kind of lecture which might be understood by some of the audience.

    The Langrangian in question is obtained by doing the following. We write down the Dirac equation:

    \(\partial_t \psi + \alpha_i \partial_i \psi = \beta M \psi\)

    Here sub-\(i\) is just \(x\), \(y\) or \(z\). Now we simply moving everthing off to the left hand side

    \(\partial_t \psi + \alpha_i \partial_i \psi - \beta M \psi = 0\)

    In variational calculus, we may want to vary the equation with another quantity. The quantity we will want to vary the equation with is \(\psi^{\dagger}\) and then we have our Langrangian for the Dirac Equation simply enough as:

    \(\psi^{\dagger}(\partial_t \psi + \alpha_i \partial_i \psi - \beta M \psi) = \mathcal{L}\)

    It is still equal to zero, but it is the Langrangian which has been varied in respect to \(\psi^{\dagger}\). This is covariant Language when we speak about this stuff, because a series of operations are known which produce new concepts we can consider. It helps describe how to move from one point point in space to another point in space. However the covariant language says that if you take \(\psi^{\dagger}\beta = \bar{\psi}\). Rewriting this in covariant language then might be a little complicated, but for now let us realize that we have successfuly reached the Dirac Langrangian, hopefully with some ease. So it might be wise, to now move onto another subject.

    Problems in Quantum Field Theory

    There are still a remaining number of problems in quantum field theory. One problem lies in it's unification. We don't know how to treat General Relativity into a unified framework with the postulates of quantum mechanics and so a unified quantum field theory still elludes us as well.

    We are not sure how to properly treat the equations of General Relativity when quantizing our objects, the question of whether the Gravitational Field for instance has a quantized fluctuation called a Graviton. A Graviton is a boson particle, just like the photon, but it's a hyothetical particle nonetheless. It has not yet been observed in nature, along with a similar problem, called Gravitational Waves. Gravitational Waves move at lightspeed and are caused by rippling over spacetime. According to General Relativity, bodies with mass accelerating should emit energy away in the form of gravitational waves (in an over-simplification).

    Niether of these gravitational phenoma have been observed to date, which means that gravity might be a psuedoforce, it will not require any physical mediator. Which means that gravity will not need to be described under any quantum field description. The problem of gravity is a huge one with many questions around it. Even Einstein, the father of the modern understanding of gravity failed to find a unification, so simply by this measure it is a massive and difficult task.

    Some believe, to do so, we must have a new reformulation of Relativity Theory, that it will take ''a new Einstein'' so to say when the final theory has been settled upon. This might be true. The fact we have not been able to unify both theories is probably an indication there is something missing from the larger picture. To our frustration however, we have searched but none of which have yielded any models which can be universally-agreed upon. We have had some other contestants beyond Einstein, such as Anthony Garret Lisi, an enthusiastic surfer who put forward a most grand theory.
    It involved very complex geometrical structures where each point on the structure corresponded to a particle in the standard model. He named his thesis ''An Exceptionally Simple Theory of Everything'' and it was an over-night hit with the press. He reached fame almost instantaneously as his theory was being evaluated by the top physicists of the day.

    His theory was remarkable, in that it made use of the geometry of what has been called, ''the most beautiful mathematical object ever.'' It is a special Gauge theory under what is called an \(E_8\). One thing it requires is the Higgs Boson, which has been of great interest the past few months due to some speculation as to whether there has been a glimpse of the particle at the LHC. In fact the Higgs Boson is part of the Standard Model as we kow it, so failure to find it will result in a remodification of how we understand matter.

    The Higgs Boson is said to be the particle at a number of possible energy ranges (where some have been excluded now from experimentation just a few weeks back), but if it is not found at all, then it means we cannot describe how an electron for instance, get's a mass.

    We seem to have believed that for a particle to get a mass, we'd require a mediator called the Higgs Boson, which provides about 1% of all the matter in the universe with mass. It even gives itself mass. We usually speculate about mass as being resultant from a fluctuation away from some ground state origin in a Mexican Hat Potential.

    Any particle which runs about the ground state of this potential has a special name, called a Goldstone boson which is analogous to a ground state photon. If your system moves away from the origin, then this costs energy to do this. The energy which is called our flucuation from the origin now has a mass and our Goldstone Boson has disappeared, but where did it go? Well the answer can be surprising, it's been ''gobbled up'' by our system. So a small movement away from the origin of this reasonably simply potential results in the description of the system possessing a mass.

    This is the Higgs phenomena in a nutshell, but I will be talking more about it in the Chapter titled ''The Higgs Boson''. Mass is therefore an unanswered question in field theory and most theories under field theory require the Higgs Boson for unification of our Standard Model. Also, in this book I will raise the question of whether quantum field theory and the unification of the equations of physics will require some description of consciousness, but for now, I am going to end that for this chapter.
    [1] http://www.nature.com/news/quantum-theorem-shakes-foundations-1.9392
    [2] http://www.newscientist.com/article/dn18669-first-quantum-effects-seen-in-visible-object.html
    [3] http://en.wikipedia.org/wiki/Copenhagen_interpretation
    [4] There is such a thing in quantum field theory called second quantization. This is when we use two special operators which act as ladders when counting ''occupation numbers in space''. Occupation numbers is the amount of particles occupying some space. Counting occupation numbers using second quantization methods is nothing but a mathematical trick but a neat one no less. Since I had been talking about the quantum Harmonic Oscillator, let us take the standard equation taught in all textbooks as
    \(H = \hbar \omega ((a^{\dagger}a) + \frac{1}{2})\)
    The creation operator is \(a^{\dagger}\) and the annihilation operator is \(a\). The first raises your quantum occupation number and the other lowers it. Using Dirac Notation to explain the way they look, the creation and annihilation operators are computed using
    \(a^{\dagger}|n> = |n+1>\sqrt{n+1}\)
    \(a|n> = |n - 1>\sqrt{n}\)
    [5] http://en.wikipedia.org/wiki/Lorentz_invariant
    [6] A. N. Vasil'ev The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics (Routledge Chapman & Hall 2004); ISBN-10: 0415310024; ISBN-13: 978-0415310024
    [7] Natural units is a system of measurement which allows us to omit a certain class of constants such as \(\hbar\), G, c by setting them to equal 1.
     
    Last edited: Dec 31, 2011
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  5. Reiku Banned Banned

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    11,238
    And AN, I am not replying to any of your posts from now on until you retract your statements here http://www.sciforums.com/showthread.php?t=111600 - which accuses me of plagairism by linking to a page written by me. I find it dishonest you haven't fixed that for future reference.
     
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  7. Reiku Banned Banned

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    Sorry, I was reading this back and noticed that the energy-momentum formula has a typo. Instead of \((Mc)^2\) it should be \(M^2 c^4\).

    I'll fix it
     
  8. Reiku Banned Banned

    Messages:
    11,238
    and those decomposed equations had extra squared terms as well. Sorry, fixed as well! Lost in the haze of latex.
     
  9. Watcher Just another old creaker Registered Senior Member

    Messages:
    366
    Your goal in your words:

    "I am still grapping with how much math and how much literature should balance for the perfect novice book." You say your audience is "...slightly comfortable with math, and within the age group of 16 and onwards... or to a very very bright 14 year old".

    But after a reasonably novice-oriented introductory section , you jump into material on the mathematics of spin, including:

    Canonical commutation relations
    Antisymmetric Levi-Cevita Symbols
    Poisson Brackets
    Hermitian conjugate transpose

    This is so discontinuous and off the wall that I really struggle to believe that you are being serious, since these concepts are beyond the comprehension of almost every 16 year old physics student. So I would have to say that you have failed to meet your original goal in a dramatic fashion.

    Interesting hobby though.
     
  10. Reiku Banned Banned

    Messages:
    11,238
    There can only be one response.

    You have not read the entire book. Because of this your statement tells me you are unaware of applications I have given in this work. One application is that subjects in general will not be attempted to be washed away with simple language. There is a guide developed for this book especially. This guide attempts to explain this stuff.

    but thanks for your comments...
     
  11. AlphaNumeric Fully ionized Registered Senior Member

    Messages:
    6,699
    You have clearly lifted material from sources without understanding it and you parrot it back. You might not be literally copying and pasting but instead you're attempting to reshuffle things so make it seem like you understand it. If not a form of plagiarism (it's a matter of debate) it's certainly dishonest. You commented about how you worry such accusations might hinder future physics jobs you might want to apply for. I can assure you if someone with a physics lecturing position considering hiring you were reading your forum posts my comments about you would be the least of your problems. In that regard you're your own worst enemy. You've done it before, you did it in that thread and you're doing it in this one. Your most recent lengthy post contains numerous pieces of evidence you don't understand what you're typing. Want a few examples?

    1) Your definitions of left and right moving waves are wrong. Not just "Oh, the definition is slightly different" but regardless of what they are referring to they are nonsensical and/or pointless. Can you tell me why?

    2) You equate scalars and matrices. Can you tell me where?

    3) You talk about the Clifford algebra and give matrix representations of the \(\alpha,\beta\) terms as if those are the matrix expressions. That isn't true. There are numerous (infinitely many even) representations of the Clifford algebra \(Cl_{1,3}\) and within each representation infinitely many different matrix expressions. Can you tell me why?

    The three mistakes I've given examples of are basic errors you'd expect someone to make if they haven't worked through explicit calculations involving the objects in question or are trying to interpret notation beyond their experience. I used to see students do it all the time and you can spot the ones trying to bluff their way through an answer a mile away.

    As Watcher says, if you think what you've written is appropriate for 16 year olds you're just in denial. The subject matter, ie Dirac's work and its place in physics, is fine but the level of detail is clearly inappropriate. You're obviously struggling to write it properly so that clearly shows it's not something a 16 year old is going to get off the bat. Quite why you persist in wasting your time like this I don't know. Ignoring my corrections and advice is only going to waste more of your own time. If you'd listened to me years ago you might actually be in a position to have a working understanding of quantum mechanics now. Instead you're still trying to deceive people into thinking you're capable.

    As for your bit about Lisi the way you write it makes it sound like no one had ever considered using \(E_{8}\) before. It was well studied in both string theory and GUT models decades before Lisi did his work (which turned out to have major problems). So saying his work was 'remarkable' for using it has a weasel word (as Wikipedia would call it). Using a much much larger gauge group in a field theory is precisely what many GUT researchers have been doing since the 70s! Furthermore, the way you phrase the bit about how it requires the Higgs particle could be misinterpreted as implying Lisi predicted it, which obviously isn't the case.

    You spend ages going into the unnecessary details about the Dirac equation then you skip over the details of the Higgs mechanism or explaining what 'gobbled up' means. It gives the distinct impression you think you've got a description of the Dirac equation's mathematics in your 'own' words down enough that you want to post it but to pad it out you put in some blurb before and after. So to add to Watcher's comments again, you not only shift violently from superficial overview to specific details you also fail to cover similar complexity subjects to the same depth. Even if your post were free of errors and you had a working understanding of all of this the 'chapter' is terribly written, it has little or no proper pacing, consideration of the readers or a clear coherent narrative with an aim/goal. It reads like a brain dump of material you can't remember, haven't planned and haven't proof read.

    Happy new year.
     
  12. Reiku Banned Banned

    Messages:
    11,238
    ''1) Your definitions of left and right moving waves are wrong. Not just "Oh, the definition is slightly different" but regardless of what they are referring to they are nonsensical and/or pointless. Can you tell me why?''

    By a sign by L and R in \(\psi\) by any chance, I get confused with notation sometimes?
     
  13. Reiku Banned Banned

    Messages:
    11,238
    ''2) You equate scalars and matrices. Can you tell me where?''

    When I say the coupled equation makes perhaps a column vector?
     
  14. Reiku Banned Banned

    Messages:
    11,238
    ''As for your bit about Lisi the way you write it makes it sound like no one had ever considered using before. It was well studied in both string theory and GUT models decades before Lisi did his work (which turned out to have major problems). So saying his work was 'remarkable' for using it has a weasel word (as Wikipedia would call it). Using a much much larger gauge group in a field theory is precisely what many GUT researchers have been doing since the 70s! Furthermore, the way you phrase the bit about how it requires the Higgs particle could be misinterpreted as implying Lisi predicted it, which obviously isn't the case.
    ''


    Well, I said at least his approach was a unified theory for quantum field theory. At no point did I distinguish between a contributor as a mathematian or a physicist.
     
  15. AlphaNumeric Fully ionized Registered Senior Member

    Messages:
    6,699
    No, getting the signs the right way around is practically a matter of convention. Look again.

    No. Want another go? Remember, I said scalar and matrix.

    It wasn't a viable unified theory and where did I say anything about mathematican vs physicist. My point was that he didn't pluck \(E_{8}\) out of mathematical obscurity and say "How about using this gauge group?", he used a group which was common place in physics. In fact one of the 5 supersymmetric string theories has \(E_{8} \times E_{8}\) as its gauge group. There's the \(E_{8}\)MSSM, the \(E_{8}\) applied to the Minimal Supersymmetric Standard Model. The decomposition of \(E_{8}\) into its subgroups, all the way down to the Standard Model's \(SU(3) \times SU(2) \times U(1)\) has been known for decades. You used the phrase "His theory was remarkable, in that it made use of the geometry of what has been called, ''the most beautiful mathematical object ever.'' ", implying that the reason it was remarkable was it used \(E_{8}\). He wasn't the first person to consider the group, not by 30 years and hundreds, even thousands, of papers. Hardly 'remarkable' on those grounds then. And calling it "'the most beautiful mathematical object ever." is a matter of debate. Who called it that? Citation needed.

    Oh and I've just noticed in your references at the bottom of the post you mention second quantisation. Your definition is wrong, utterly wrong. What you've given is what you get after first quantisation. Basic ladder operators come up in basic quantum mechanics, they are one of the first things you learn when doing bra-ket notation. You've just demonstrated, again, you aren't familiar with this stuff and you're presenting yourself as knowledgeable when you are not. The quantum harmonic oscillator you give is a nice example of a first quantised system. Second quantisation involves path integrals and unpleasant measures on the fields.

    By the time someone gets to the Dirac equation they should understand the difference between the first quantisation of standard quantum mechanics and the more advanced methods of second quantisation in quantum field theory. You clearly have not got that understanding and in this case it's led to you saying things demonstrably wrong. This is the whole point of my complaints about you doing this sort of thread, you don't understand really what you're posting, you're just patching together little snips from different sources in the same way someone might try to construct a sentence in French by looking up the individual words in a language dictionary. Just as it leads to incorrect translations and obvious nonsense to those who understand French your endeavour is riddled with such errors. In this case you made the mistake of not checking Wikipedia before hand and just said it. Or perhaps you did and you saw a lot of similar looking symbols and thought you were saying something correct. That's the problem with not understanding the symbols and just playing character matching.

    It's one thing for you to post "Hey, I've got a pet theory, wanna hear it?" threads but now you're deliberately aiming at people who are not sufficiently familiar with the subject matter to spot your mistakes, which you are also unable to spot for the same reason. I get it, you want to convince people you've capable at physics and now you've realised the only people you can convince are those less informed than you but it's extremely dishonest and almost despicable to be willing to poison someone else's understanding in order to delude yourself.
     
  16. Reiku Banned Banned

    Messages:
    11,238
    Look alphanumeric, if this is what I wrote, then I am under the impression it is right. Now unless you have any reason to think something is wrong and you want to make that known, then please do tell. This was the main reason I came here was it not, to rectify problems with the work.

    Now, I don't know how I have messed up my right mover-left mover definitions, nor am I sure wew I have messed up equating a scalar with a matrix. I don't think it has anything to do with the spin matrices, but who knows? Nothing is popping out at me. So nip it in the bud alphenumeric if you feel you have something to say. I can get other people to look at the work, I'd never need to just fall on this place.
     
  17. Reiku Banned Banned

    Messages:
    11,238
    As for Lisi, I was quite aware the mathematical object existed before him, but I can see how that line could mean something else to someone else.
     
  18. steampunk Registered Senior Member

    Messages:
    278
    I don't think they teach the prerequisites for this stuff that's assumed to be known in your book. Not for this age group. I personally got the impression that quantum mechanics as more of a religion than a science, so I never pursued it seriously. I'm comfortable with the physics that puts satellites into space, Newtonian physics. Perhaps a conceptual book, like Paul Hewitt wrote would be good for 16, even 13 year olds.

    Could you convince what's practical about qm? Why should students understand this stuff? How does it benefit them? When they know it, how does it prepare them for a money making opportunity? How is this knowledge useful in the marketplace? What kind of commitment are they looking at in terms of full understanding of it. Such an explanation could help a student understand if qm is for them.
     
  19. AlphaNumeric Fully ionized Registered Senior Member

    Messages:
    6,699
    But you don't know the material properly and you didn't check, yet you're here trying to present yourself as knowledgeable, trying to 'teach' people who don't know you're mistaken.

    It's dishonest.

    Hasn't the last 4 or 5 years of multiple people pointing out mistake after mistake after mistake of yours given you a clue?

    Oh please, you wouldn't be whining so much if you were really after corrections. You would have checked your facts before hand. You are ill prepared to write anything of this sort, trying to teach people things you don't know. You are doing it to appear knowledgeable and it's why you aren't allowed in the main physics forum, we're sick of it.

    Did you even bother to read your expressions? You wrote \(\frac{\partial \psi_{L}}{\partial t} = +\frac{\partial \psi_{L}}{\partial t}\). That's a tautology, it doesn't imply anything. You also wrote \(\frac{\partial \psi_{R}}{\partial t} = -\frac{\partial \psi_{R}}{\partial t}\). That implies \(\frac{\partial \psi_{R}}{\partial t} = 0\).

    A wave is dependent on the combination \(x \pm vt\) so that if \(\psi\) is a wave then it will satisfy \(\frac{\partial \psi}{\partial t} = \pm v \frac{\partial \psi}{\partial x}\). See the difference? It's a relationship between t and x derivatives, you forgot the x derivative and didn't notice it even when I pointed you to them.

    As for vectors and scalars, you write \(\omega = \alpha k + M\). M is a scalar, k and \(\omega\) are vectors.

    Those are basic things a 1st year would pick up on. The more conceptual stuff, which someone familiar with the Dirac equation, spin matrices etc should know, is that your statements about the form of the \(\alpha,\beta\) matrices are wrong. You give explicit forms of the \(Cl_{1,1}\) algebra matrices and \(Cl_{1,3}\) algebra matrices and imply they are the forms of the matrices. No, they are representations. For example, there's the chiral representation and the Weyl representation commonly used for the \(Cl_{1,3}\) matrices in the usual Dirac equation. There's infinitely many different sets of matrices which satisfy those equations, for example changing coordinates. Physicists pick the representation must convenient for their purposes. This is something you should know if you have a decent understanding of the Dirac equation, something you pick up by working with it. This, once again, demonstrates you don't have anywhere near the grasp you think you do. You have little more than an understanding which can be gained from reading Wikipedia for a weekend.

    Seriously? You're asking what applications quantum mechanics has in the real world?

    Nuclear power, nuclear weapons, the CPUs all computers work on, scanning electron microscopes, MRI and PET scanners for doctors, protein folding for cancer research, superconductors for sensitive transmitter/detectors in mobile networks, new materials using nanotechnology where quantum processes are used in the construction, lasers, solar cells.

    All of those have or soon will reshape areas of our civilization. What would the world be like if we didn't have computers? Or lasers? Or nuclear anything? If doctors had to cut you open to see inside you? If we wouldn't examine virus sheaths and individual cells so closely we can make out individual molecules?

    A grasp of quantum mechanics has been at the core of much of the advancement in the 20th century and as computers are now bringing molecular quantum dynamics into practical reach it'll only accelerate.
     
  20. Reiku Banned Banned

    Messages:
    11,238
    You think I understand wiki? If you think that is my source of information I dare say you are barking up the wrong tree. Wiki tends to use a lot of notation I am unfamiliar with... A bit like yourself recently with farsight, just as an example.

    Anyway... you have me confused, help me clear my confusion. This is why I am confused: Let us now revert to not using natural units... to make this clearer...

    The Special Relativity 4-momentum for any particle satisfies \(E^2-p^2c^2=m^2c^4\). From quantum mechanics we can state that \(E=\hbar \omega\) is energy and \(p=\hbar k\). This leads to the realization that \(\omega^2-k^2c^2=\frac{m^2c^4}{ \hbar^2}\). Then one can recover the dispersion relation \(\omega=\sqrt{(\frac{k^2c^2+m^2c^4}{\hbar^2})}\).

    For those not following, \(\omega=\sqrt{(\frac{k^2c^2+m^2c^4}{\hbar^2})}\) is the same as \(\omega = k + M\) in natural units....


    So were have I gone wrong exactly?
     
  21. AlphaNumeric Fully ionized Registered Senior Member

    Messages:
    6,699
    I use notation common to textbooks and lecture courses. If you or Farsight do not understand things expected of 1st year undergraduates the problem is with you, not anyone else.

    Thank you so much for that because not only have you demonstrated you don't get the energy-mass-momentum expression properly but you've made an algebraic mistake so horrifically bad 14 year olds should be ashamed to make it.

    I'll start with the equation issue. The 4-momentum expression is \(p_{\mu}p^{\mu} = -M^{2}\) (for the -+++ signature), which expands to \(-E^{2} + \mathbf{p}\cdot \mathbf{p} = -M^{2}\). Momentum is a VECTOR, so to combine the scalar quantity \(E^{2}\) to it in some way you have to use the dot product (or some other vector to scalar map). If space has 3 dimensions so \(\mathbf{p} = (p_{1},p_{2},p_{3})\) the expression then becomes \(-E^{2} + p_{1}^{2} + p_{2}^{2} + p_{3}^{2} = -M^{2}\). Clearly you couldn't write down \(E + \mathbf{p} = M\) because it would be nonsense, regardless of the physical meaning of each term. The only way that could even have a chance to be right is if you work in 1+1 dimensional space-time so \(\mathbf{p} = p\), a scalar.

    But even in that case you don't end up with the expression you claim because of your massive terrible mistake. You have said \(\omega = \sqrt{k^{2}+M^{2}} = k + M\). Even allowing for k being a scalar, when its actually a vector, you've said \(\sqrt{a^{2}+b^{2}} = a+b\). That's utterly wrong as an identity and it's a mistake you really shouldn't be making. It's a mistake children shouldn't be making. It's easy to see why it is false, just square both sides, \(a^{2}+b^{2} = (a+b)^{2} = a^{2}+b^{2} + 2ab\) so it's only true if \(2ab = 0\), either a=0 or b=0 or both. Try it with numbers, put in a=b=1. You just claimed \(\sqrt{1^{2}+1^{2}} = 1+1\) but that's saying \(\sqrt{2}=2\).

    It's clear what you've tried to do. You've taken a formula from somewhere, be it Wiki or some other website, and you've attempted to do something with it, in this case what you think is a simplification. But in doing that you've shown how little algebra you really understand because you did it incorrectly.

    And before you try to make excuses about now working with this etc, you've previously claimed to have done sufficiently advanced general relativity at college to be covering curvature. If you really were familiar with such stuff, you'd have seen expressions like \(ds =\sqrt{dx^{2} + dy^{2}}\) for the Euclidean 2-norm, which are not simplified down to \(ds = |dx| + |dy|\). If such simplifications were true then all of the p-norms, \(ds = ( \sum dx_{i}^p )^{\frac{1}{p}}\) would be equal via \(ds = ( \sum dx_{i}^p )^{\frac{1}{p}} = \sum (dx_{i}^{p})^{\frac{1}{p}} = \sum |dx_{i}|\). Only the 1-norm is equal to that last expression. So if you weren't lying about doing such things in college these principles should be drilled into you. Of course it's unquestionable that your claim to be doing in pre-university college that stuff is a lie. No pre-university educational establishment teaches tensor calculus on pseudo-Riemannian manifolds, in fact studying the specifics of black hole curvature properties is something Cambridge students only really get into in their masters year. Remember how you were saying in the open government forum about how you're so honest? I guess you weren't including that claim of yours in your evaluation of yourself.
     
  22. Reiku Banned Banned

    Messages:
    11,238
    I knew you were going to equate the two. Notice I didn't?

    I said they were the same, in that you can rearrange the components of the equation. If I said the two were equal, I would have done what you did above, but I didn't.

    You are also hiding behind this smokescreen quite aware that you have forgotten I was using natural units, because you said

    ''M is a scalar, k and omega are vectors.''

    Well, yeah, M on it's own is a scalar. But \(Mc^2\) or rather the form \(M^2c^4\) is a vector. So clinging onto the alleged criticism that I don't know what I am doing, you are creating equations above which I have never stated and not even admitting that you messed that up concerning the equation not fitting physics.

    Nice one.
     
  23. Reiku Banned Banned

    Messages:
    11,238
    Why do you do this all the time, I mean misrepresenting what I mean? If I am unclear, could you simply not just ask instead of making vacuous assumptions?

    One can clearly see this is what I meant, by not equating the two, but you did. Note ''equate'' is the key word here as well. When I say ''the same'' I don't mean to equate the two equations.

    For instance, I said that equation was the same as \(\omega = k + M\).. since it is more than clear you messed up forgetting we were talking in natural units, the dirac equation is

    \(i\hbar \partial_t \psi = i c\hbar \alpha \cdot \nabla \psi + \beta Mc^2 \psi\)
    Obviously not using natural units. Turning this into the form of the equation \(\omega = k + M\) as

    \(i\hbar \partial_t \psi = i ck \alpha \cdot \nabla \psi + \beta Mc^2 \psi\)

    Thus one can see that the sign ''talking the square root'' of k^2 +M^2 does not appear in the dirac equation. In fact, if it did, the equation would be rather cumbersome.
     

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