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| In [[quantum field theory]], the '''Dirac spinor''' is the [[bispinor]] in the [[Plane wave|plane-wave]] solution
| | Creating Meals Storage For Uncertain Times<br><br>Have you at any time been out of work for a week, a thirty day period, even more time? Do you wish you could buy foodstuff ONLY when they are on sale and with coupon codes? Do you wish you had a stockpile of grains now that the charges are acquiring so higher?<br><br>Another excellent rule of thumb is to pay out consideration to the foodstuff on the outside shelves of the store. This is in which the more healthy things [https://www.diigo.com/user/Ebontalifarro Ebon Talifarro] like dairy, make, seafood, and meats are retained.<br><br>One of the blunders homeowners commit is putting in one particular-be aware lights. They are way too scared to go outdoors the box and try mixing different varieties of light. Lighting is a good deal like style. You have to blend and match to locate the very best ensemble. Most folks stick with recessed lights simply because it is safer. But if you are right after present day techniques of light creating and you want to intensify the elegance of your kitchen, steer clear of recessed lights.<br><br>To shut the container, just press the outer body down onto Ebon Talifarro the lid to have interaction the inner locking tabs and silicone ring for an instant, spill proof seal. The container bodies nest inside of each other for simple storage and stacking.<br><br>Leftover meats can be frozen for two to a few months although complete, raw cuts of meat can be frozen for up to a yr. To avoid freezer burn up, wrap parts of meat in possibly foil and/or plastic wrap numerous moments and shop in an air limited container or bag.<br><br>13. Skinny girls will select the vegetarian option when accessible. Not all skinny women are vegetarian, but research present substituting greens for meat dishes each now and once again can assist lower back on calories and boost digestion.<br><br>Your prolonged time period Ebon Talifarro is composed of obtaining a 12 months's source really worth of life-sustaining food items that have a lengthy shelf-existence. You possibly received't be rotating by means of this foods as a lot given that it will be items these kinds of as wheat, white rice, dried beans, powdered milk, and many others. But because the shelf lifestyle is so extended you can gradually buy the things when they are on sale and work up to a year's provide. If you only have to change some things following 10, fifteen, 20 many years it will not be a massive damper on your month-to-month spending budget. If you get courageous sufficient to start utilizing your lengthy expression meals storage items you can preserve some money in the brief time period.<br><br>In the extensive greater part of new houses, the builder puts in mounted wire [http://Www.Bing.com/search?q=shelves&form=MSNNWS&mkt=en-us&pq=shelves shelves] throughout. Not only does the wire shelving allow small products (now in which did that soup mix go?) to fall through, but they cannot be altered to your needs. Standard storage panels are on the 32mm (one.twenty five") European normal and as a result shelves can be adjusted at that increment up or down.<br><br>Small and regular progress equals lengthy phrase success. So start off out with 1 tiny area of your home or one particular region of your office versus making an attempt to do also considerably all at once. You'll see outcomes Ebon Talifarro rapidly and be able to maintain your recently sorted spot a lot more effectively. |
| :<math>\psi = \omega_\vec{p}\;e^{-ipx} \;</math>
| |
| of the free [[Dirac equation]], | |
| :<math>(i\gamma^\mu\partial_{\mu}-m)\psi=0 \;,</math>
| |
| where (in the units <math>\scriptstyle c \,=\, \hbar \,=\, 1</math>)
| |
| :<math>\scriptstyle\psi</math> is a [[Theory of relativity|relativistic]] [[spin-1/2]] [[Field (physics)|field]],
| |
| :<math>\scriptstyle\omega_\vec{p}</math> is the Dirac [[spinor]] related to a plane-wave with [[wave-vector]] <math>\scriptstyle\vec{p}</math>,
| |
| :<math>\scriptstyle px \;\equiv\; p_\mu x^\mu</math>,
| |
| :<math>\scriptstyle p^\mu \;=\; \{\pm\sqrt{m^2+\vec{p}^2},\, \vec{p}\}</math> is the four-wave-vector of the plane wave, where <math>\scriptstyle\vec{p}</math> is arbitrary,
| |
| :<math>\scriptstyle x^\mu</math> are the four-coordinates in a given [[inertial frame]] of reference.
| |
| | |
| The Dirac spinor for the positive-frequency solution can be written as
| |
| :<math> | |
| \omega_\vec{p} =
| |
| \begin{bmatrix}
| |
| \phi \\ \frac{\vec{\sigma}\vec{p}}{E_{\vec{p}} + m} \phi
| |
| \end{bmatrix} \;,
| |
| </math>
| |
| where
| |
| :<math>\scriptstyle\phi</math> is an arbitrary two-spinor,
| |
| :<math>\scriptstyle\vec{\sigma}</math> are the [[Pauli matrices]],
| |
| :<math>\scriptstyle E_\vec{p}</math> is the positive square root <math>\scriptstyle E_{\vec{p}} \;=\; +\sqrt{m^2+\vec{p}^2}</math>
| |
| | |
| ==Derivation from Dirac equation==
| |
| The Dirac equation has the form
| |
| :<math>\left(-i \vec{\alpha} \cdot \vec{\nabla} + \beta m \right) \psi = i \frac{\partial \psi}{\partial t} \,</math>
| |
| | |
| In order to derive the form of the four-spinor <math>\scriptstyle\omega</math> we have to first note the value of the matrices α and β:
| |
| :<math>\vec\alpha = \begin{bmatrix} \mathbf{0} & \vec{\sigma} \\ \vec{\sigma} & \mathbf{0} \end{bmatrix} \quad \quad \beta = \begin{bmatrix} \mathbf{I} & \mathbf{0} \\ \mathbf{0} & -\mathbf{I} \end{bmatrix} \,</math>
| |
| These two 4×4 matrices are related to the [[Gamma matrices|Dirac gamma matrices]]. Note that '''0''' and '''I''' are 2×2 matrices here.
| |
| | |
| The next step is to look for solutions of the form
| |
| :<math>\psi = \omega e^{-i p \cdot x}</math>,
| |
| while at the same time splitting ω into two two-spinors:
| |
| :<math>\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,</math>.
| |
| | |
| ===Results===
| |
| Using all of the above information to plug into the Dirac equation results in
| |
| :<math>E \begin{bmatrix} \phi \\ \chi \end{bmatrix} =
| |
| \begin{bmatrix} m \mathbf{I} & \vec{\sigma}\vec{p} \\ \vec{\sigma}\vec{p} & -m \mathbf{I} \end{bmatrix} \begin{bmatrix} \phi \\ \chi \end{bmatrix} \,</math>.
| |
| | |
| This matrix equation is really two coupled equations:
| |
| :<math>\left(E - m \right) \phi = \left(\vec{\sigma}\vec{p} \right) \chi \,</math>
| |
| :<math>\left(E + m \right) \chi = \left(\vec{\sigma}\vec{p} \right) \phi \,</math>
| |
| | |
| Solve the 2nd equation for <math>\scriptstyle \chi \,</math> and one obtains
| |
| :<math>\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} \phi \\ \frac{\vec{\sigma}\vec{p}}{E + m} \phi \end{bmatrix} \,</math>.
| |
| | |
| Solve the 1st equation for <math>\phi \,</math> and one finds
| |
| :<math>\omega = \begin{bmatrix} \phi \\ \chi \end{bmatrix} = \begin{bmatrix} - \frac{\vec{\sigma}\vec{p}}{-E + m} \chi \\ \chi \end{bmatrix} \,</math>.
| |
| This solution is useful for showing the relation between [[anti-particle]] and particle.
| |
| | |
| ==Details==
| |
| ===Two-spinors===
| |
| The most convenient definitions for the two-spinors are:
| |
| :<math>\phi^1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \quad \quad \phi^2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \,</math>
| |
| | |
| and
| |
| :<math>\chi^1 = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad \quad \chi^2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \,</math>
| |
| | |
| ===Pauli matrices===
| |
| The [[Pauli matrices]] are
| |
| :<math>
| |
| \sigma_1 =
| |
| \begin{bmatrix}
| |
| 0&1\\
| |
| 1&0
| |
| \end{bmatrix}
| |
| \quad \quad
| |
| \sigma_2 =
| |
| \begin{bmatrix}
| |
| 0&-i\\
| |
| i&0
| |
| \end{bmatrix}
| |
| \quad \quad
| |
| \sigma_3 =
| |
| \begin{bmatrix}
| |
| 1&0\\
| |
| 0&-1
| |
| \end{bmatrix}
| |
| </math>
| |
| | |
| Using these, one can calculate:
| |
| :<math>\vec{\sigma}\vec{p} = \sigma_1 p_1 + \sigma_2 p_2 + \sigma_3 p_3 =
| |
| \begin{bmatrix}
| |
| p_3 & p_1 - i p_2 \\
| |
| p_1 + i p_2 & - p_3
| |
| \end{bmatrix}</math>
| |
| | |
| ==Four-spinor for particles==
| |
| Particles are defined as having ''positive'' energy. The normalization for the four-spinor ω is chosen so that <math>\scriptstyle\omega^\dagger \omega \;=\; 2 E \,</math> {{Elucidate|date=February 2012}}. These spinors are denoted as ''u'':
| |
| | |
| :<math> u(\vec{p}, s) = \sqrt{E+m}
| |
| \begin{bmatrix}
| |
| \phi^{(s)}\\
| |
| \frac{\vec{\sigma} \cdot \vec{p} }{E+m} \phi^{(s)}
| |
| \end{bmatrix} \,</math>
| |
| where ''s'' = 1 or 2 (spin "up" or "down")
| |
| | |
| Explicitly,
| |
| :<math>u(\vec{p}, 1) = \sqrt{E+m} \begin{bmatrix}
| |
| 1\\
| |
| 0\\
| |
| \frac{p_3}{E+m} \\
| |
| \frac{p_1 + i p_2}{E+m}
| |
| \end{bmatrix} \quad \mathrm{and} \quad
| |
| u(\vec{p}, 2) = \sqrt{E+m} \begin{bmatrix}
| |
| 0\\
| |
| 1\\
| |
| \frac{p_1 - i p_2}{E+m} \\
| |
| \frac{-p_3}{E+m}
| |
| \end{bmatrix} </math>
| |
| | |
| ==Four-spinor for anti-particles==
| |
| Anti-particles having ''positive'' energy <math>\scriptstyle E</math> are defined as particles having ''negative'' energy and propagating backward in time. Hence changing the sign of <math>\scriptstyle E</math> and <math>\scriptstyle \vec{p}</math> in the four-spinor for particles will give the four-spinor for anti-particles:
| |
| | |
| :<math> v(\vec{p},s) = \sqrt{E+m}
| |
| \begin{bmatrix}
| |
| \frac{\vec{\sigma} \cdot \vec{p} }{E+m} \chi^{(s)}\\
| |
| \chi^{(s)}
| |
| \end{bmatrix} \,</math>
| |
| | |
| Here we choose the <math>\scriptstyle\chi</math> solutions. Explicitly,
| |
| :<math>v(\vec{p}, 1) = \sqrt{E+m} \begin{bmatrix}
| |
| \frac{p_1 - i p_2}{E+m} \\
| |
| \frac{-p_3}{E+m} \\
| |
| 0\\
| |
| 1
| |
| \end{bmatrix} \quad \mathrm{and} \quad
| |
| v(\vec{p}, 2) = \sqrt{E+m} \begin{bmatrix}
| |
| \frac{p_3}{E+m} \\
| |
| \frac{p_1 + i p_2}{E+m} \\
| |
| 1\\
| |
| 0\\
| |
| \end{bmatrix} </math>
| |
| | |
| ==Completeness relations==
| |
| The completeness relations for the four-spinors ''u'' and ''v'' are
| |
| :<math>\sum_{s=1,2}{u^{(s)}_p \bar{u}^{(s)}_p} = p\!\!\!/ + m \,</math>
| |
| :<math>\sum_{s=1,2}{v^{(s)}_p \bar{v}^{(s)}_p} = p\!\!\!/ - m \,</math>
| |
| | |
| where
| |
| :<math>p\!\!\!/ = \gamma^\mu p_\mu \,</math> (see [[Feynman slash notation#With four-momentum|Feynman slash notation]])
| |
| :<math>\bar{u} = u^{\dagger} \gamma^0 \,</math>
| |
| | |
| ==Dirac spinors and the Dirac algebra==
| |
| The [[Dirac matrices]] are a set of four 4×4 [[Matrix (mathematics)|matrices]] that are used as [[Spin (physics)|spin]] and [[Charge (physics)|charge]] [[Operator (physics)|operators]].
| |
| | |
| ===Conventions===
| |
| There are several choices of [[Signature (physics)|signature]] and [[Group representation|representation]] that are in common use in the physics literature. The Dirac matrices are typically written as <math>\scriptstyle \gamma^\mu</math> where <math>\scriptstyle \mu</math> runs from 0 to 3. In this notation, 0 corresponds to time, and 1 through 3 correspond to x, y, and z.
| |
| | |
| The + − − − [[Signature (physics)|signature]] is sometimes called the [[West Coast of the United States|west coast]] metric, while the − + + + is the [[East Coast of the United States|east coast]] metric. At this time the + − − − signature is in more common use, and our example will use this signature. To switch from one example to the other, multiply all <math>\scriptstyle\gamma^\mu</math> by <math>\scriptstyle i</math>.
| |
| | |
| After choosing the signature, there are many ways of constructing a representation in the 4×4 matrices, and many are in common use. In order to make this example as general as possible we will not specify a representation until the final step. At that time we will substitute in the [[Chirality (physics)|"chiral"]] or [[Hermann Weyl|"Weyl"]] representation as used in the popular graduate textbook ''An Introduction to Quantum Field Theory'' by [[Michael Peskin|Michael E. Peskin]] and [[Daniel Schroeder|Daniel V. Schroeder]].
| |
| | |
| ===Construction of Dirac spinor with a given spin direction and charge===
| |
| First we choose a [[spin (physics)|spin]] direction for our electron or positron. As with the example of the Pauli algebra discussed above, the spin direction is defined by a [[unit vector]] in 3 dimensions, (a, b, c). Following the convention of Peskin & Schroeder, the spin operator for spin in the (a, b, c) direction is defined as the dot product of (a, b, c) with the vector
| |
| :<math>(i\gamma^2\gamma^3,\;\;i\gamma^3\gamma^1,\;\;i\gamma^1\gamma^2) = -(\gamma^1,\;\gamma^2,\;\gamma^3)i\gamma^1\gamma^2\gamma^3</math>
| |
| | |
| :<math>\sigma_{(a,b,c)} = ia\gamma^2\gamma^3 + ib\gamma^3\gamma^1 + ic\gamma^1\gamma^2</math>
| |
| | |
| Note that the above is a [[root of unity]], that is, it squares to 1. Consequently, we can make a [[projection operator]] from it that projects out the sub-algebra of the Dirac algebra that has spin oriented in the (a, b, c) direction:
| |
| | |
| :<math>P_{(a,b,c)} = \frac{1}{2}\left(1 + \sigma_{(a,b,c)}\right)</math>
| |
| | |
| Now we must choose a charge, +1 (positron) or −1 (electron). Following the conventions of Peskin & Schroeder, the operator for charge is <math>\scriptstyle Q \,=\, -\gamma^0</math>, that is, electron states will take an eigenvalue of −1 with respect to this operator while positron states will take an eigenvalue of +1.
| |
| | |
| Note that <math>\scriptstyle Q</math> is also a square root of unity. Furthermore, <math>\scriptstyle Q</math> commutes with <math>\scriptstyle\sigma_{(a, b, c)}</math>. They form a [[complete set of commuting operators]] for the Dirac algebra. Continuing with our example, we look for a representation of an electron with spin in the (a, b, c) direction. Turning <math>\scriptstyle Q</math> into a projection operator for charge = −1, we have
| |
| | |
| :<math>P_{-Q} = \frac{1}{2}\left(1 - Q\right) = \frac{1}{2}\left(1 + \gamma^0\right)</math> | |
| | |
| The projection operator for the spinor we seek is therefore the product of the two projection operators we've found:
| |
| | |
| :<math>P_{(a, b, c)}\;P_{-Q}</math>
| |
| | |
| The above projection operator, when applied to any spinor, will give that part of the spinor that corresponds to the electron state we seek. So we can apply it to a spinor with the value 1 in one of its components, and 0 in the others, which gives a column of the matrix. Continuing the example, we put (a, b, c) = (0, 0, 1) and have
| |
| | |
| :<math>P_{(0, 0, 1)} = \frac{1}{2}\left(1+ i\gamma_1\gamma_2\right)</math>
| |
| | |
| and so our desired projection operator is
| |
| | |
| :<math>P = \frac{1}{2}\left(1+ i\gamma^1\gamma^2\right) \cdot \frac{1}{2}\left(1 + \gamma^0\right) =
| |
| \frac{1}{4}\left(1+\gamma^0 +i\gamma^1\gamma^2 + i\gamma^0\gamma^1\gamma^2\right)</math>
| |
| | |
| The 4×4 gamma matrices used in the Weyl representation are
| |
| | |
| :<math>\gamma_0 = \begin{bmatrix}0&1\\1&0\end{bmatrix}</math>
| |
| :<math>\gamma_k = \begin{bmatrix}0&\sigma^k\\ -\sigma^k& 0\end{bmatrix}</math>
| |
| | |
| for k = 1, 2, 3 and where <math>\sigma^i</math> are the usual 2×2 [[Pauli matrices]]. Substituting these in for P gives
| |
| | |
| :<math>P = \frac14\begin{bmatrix}1+\sigma^3&1+\sigma^3\\1+\sigma^3&1+\sigma^3\end{bmatrix}
| |
| =\frac12\begin{bmatrix}1&0&1&0\\0&0&0&0\\ 1&0&1&0\\0&0&0&0\end{bmatrix}</math>
| |
| | |
| Our answer is any non-zero column of the above matrix. The division by two is just a normalization. The first and third columns give the same result:
| |
| | |
| :<math>\left|e^-,\, +\frac12\right\rangle =
| |
| \begin{bmatrix}1\\0\\1\\0\end{bmatrix}</math>
| |
| | |
| More generally, for electrons and positrons with spin oriented in the (a, b, c) direction, the projection operator is
| |
| | |
| :<math>\frac14\begin{bmatrix}
| |
| 1+c&a-ib&\pm (1+c)&\pm(a-ib)\\
| |
| a+ib&1-c&\pm(a+ib)&\pm (1-c)\\
| |
| \pm (1+c)&\pm(a-ib)&1+c&a-ib\\
| |
| \pm(a+ib)&\pm (1-c)&a+ib&1-c
| |
| \end{bmatrix}</math>
| |
| | |
| where the upper signs are for the electron and the lower signs are for the positron. The corresponding spinor can be taken as any non zero column. Since <math>\scriptstyle a^2+b^2+c^2 \,=\, 1</math> the different columns are multiples of the same spinor. The representation of the resulting spinor in the [[Gamma matrices#Dirac basis|Dirac basis]] can be obtained using the rule given in the [[bispinor]] article.
| |
| | |
| ==See also==
| |
| *[[Dirac equation]]
| |
| *[[Helicity Basis]]
| |
| *[[Spin(3,1)]], the [[double cover]] of [[SO(3,1)]] by a [[spin group]]
| |
| | |
| ==References==
| |
| *{{cite book
| |
| | last = Aitchison
| |
| | first = I.J.R.
| |
| | authorlink =
| |
| | coauthors = A.J.G. Hey
| |
| | title = Gauge Theories in Particle Physics (3rd ed.)
| |
| | publisher = Institute of Physics Publishing
| |
| |date=September 2002
| |
| | location =
| |
| | pages =
| |
| | url =
| |
| | doi =
| |
| | isbn = 0-7503-0864-8 }}
| |
| * {{Cite web
| |
| | first = David
| |
| | last = Miller
| |
| | title = Relativistic Quantum Mechanics (RQM)
| |
| | year = 2008
| |
| | pages = 26–37
| |
| | url = http://www.physics.gla.ac.uk/~dmiller/lectures/RQM_2008.pdf
| |
| | postscript = <!--None-->
| |
| }}
| |
| | |
| [[Category:Quantum mechanics]]
| |
| [[Category:Quantum field theory]]
| |
| [[Category:Spinors]]
| |
| [[Category:Paul Dirac|Spinor]]
| |
Creating Meals Storage For Uncertain Times
Have you at any time been out of work for a week, a thirty day period, even more time? Do you wish you could buy foodstuff ONLY when they are on sale and with coupon codes? Do you wish you had a stockpile of grains now that the charges are acquiring so higher?
Another excellent rule of thumb is to pay out consideration to the foodstuff on the outside shelves of the store. This is in which the more healthy things Ebon Talifarro like dairy, make, seafood, and meats are retained.
One of the blunders homeowners commit is putting in one particular-be aware lights. They are way too scared to go outdoors the box and try mixing different varieties of light. Lighting is a good deal like style. You have to blend and match to locate the very best ensemble. Most folks stick with recessed lights simply because it is safer. But if you are right after present day techniques of light creating and you want to intensify the elegance of your kitchen, steer clear of recessed lights.
To shut the container, just press the outer body down onto Ebon Talifarro the lid to have interaction the inner locking tabs and silicone ring for an instant, spill proof seal. The container bodies nest inside of each other for simple storage and stacking.
Leftover meats can be frozen for two to a few months although complete, raw cuts of meat can be frozen for up to a yr. To avoid freezer burn up, wrap parts of meat in possibly foil and/or plastic wrap numerous moments and shop in an air limited container or bag.
13. Skinny girls will select the vegetarian option when accessible. Not all skinny women are vegetarian, but research present substituting greens for meat dishes each now and once again can assist lower back on calories and boost digestion.
Your prolonged time period Ebon Talifarro is composed of obtaining a 12 months's source really worth of life-sustaining food items that have a lengthy shelf-existence. You possibly received't be rotating by means of this foods as a lot given that it will be items these kinds of as wheat, white rice, dried beans, powdered milk, and many others. But because the shelf lifestyle is so extended you can gradually buy the things when they are on sale and work up to a year's provide. If you only have to change some things following 10, fifteen, 20 many years it will not be a massive damper on your month-to-month spending budget. If you get courageous sufficient to start utilizing your lengthy expression meals storage items you can preserve some money in the brief time period.
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Small and regular progress equals lengthy phrase success. So start off out with 1 tiny area of your home or one particular region of your office versus making an attempt to do also considerably all at once. You'll see outcomes Ebon Talifarro rapidly and be able to maintain your recently sorted spot a lot more effectively.