theories:quantum_field_theory:canonical
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theories:quantum_field_theory:canonical [2018/05/05 12:31] jakobadmin [Abstract] |
theories:quantum_field_theory:canonical [2020/01/03 02:54] (current) 110.70.51.250 [Intuitive] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
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| - | **A nice analogy** | ||
| <blockquote>The basic object is the ocean. In the context of particle physics, this ocean is then called a field. Such a field is now existing at every point in space and at every instance in time. In the very literally meaning of the word, it fills up all of the universe. If there is nothing of interest around, this is because the size of the field at this point in space and time is small or even vanishing. However, if there is a spike at some point in the field then just as in the picture of the ocean there sits a particle. If there is a second spike somewhere else, then there is another particle, and so on. Since all the spikes belong to the same field, they describe the same type of particle, say an electron. The spikes may move with different speeds, so the electrons appear to have different speeds, but they are still electrons. That is the reason why all electrons are the same: They are just spikes in the same field. Such a spike is often called an excitation of the field, and this excitation is the electron. | <blockquote>The basic object is the ocean. In the context of particle physics, this ocean is then called a field. Such a field is now existing at every point in space and at every instance in time. In the very literally meaning of the word, it fills up all of the universe. If there is nothing of interest around, this is because the size of the field at this point in space and time is small or even vanishing. However, if there is a spike at some point in the field then just as in the picture of the ocean there sits a particle. If there is a second spike somewhere else, then there is another particle, and so on. Since all the spikes belong to the same field, they describe the same type of particle, say an electron. The spikes may move with different speeds, so the electrons appear to have different speeds, but they are still electrons. That is the reason why all electrons are the same: They are just spikes in the same field. Such a spike is often called an excitation of the field, and this excitation is the electron. | ||
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| Then what is about the other types of particles? The quarks, the gluons, the Higgs? Well, these belong just to other fields. That is, our universe is filled up with many fields, all existing simultaneously at every point in space and time. | Then what is about the other types of particles? The quarks, the gluons, the Higgs? Well, these belong just to other fields. That is, our universe is filled up with many fields, all existing simultaneously at every point in space and time. | ||
| - | You may be wondering how this should work, and if this is not a bit crowded. But you know already that fields are mathematical concepts. For example, you can associate with every point in space and time a temperature, and thus create a temperature field. At the same time, there is an atmospheric pressure field. Both can happily exist simultaneously. But they are not ignoring each other. As you know, both a related with each other: If either changes this indicates a change of the other as well. Though this analogy is not exactly the same as the particle physics fields, and there are more things involved, the basic idea is the same. | + | You may be wondering how this should work, and if this is not a bit crowded. But you know already that fields are mathematical concepts. For example, you can associate with every point in space and time a temperature, and thus create a temperature field. At the same time, there is an atmospheric pressure field. Both can happily exist simultaneously. But they are not ignoring each other. As you know, both are related with each other: If either changes this indicates a change of the other as well. Though this analogy is not exactly the same as the particle physics fields, and there are more things involved, the basic idea is the same. |
| Also the particle physics fields interact, and thus not ignore each other. [...] | Also the particle physics fields interact, and thus not ignore each other. [...] | ||
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| ---- | ---- | ||
| - | Arguably the most important equation of quantum field theory is the [[equations:canonical_commutation_relations|canonical commutation relation]] | + | Arguably the most important equation of quantum field theory is the [[formulas:canonical_commutation_relations|canonical commutation relation]] |
| \begin{equation} \label{qftcomm} [\Phi(x), \pi(y)]=\Phi(x) \pi(y) - \pi(y) \Phi(x) = i \delta(x-y) \end{equation} where $\delta(x-y)$ is the Dirac delta distribution and $\pi(y) = \frac{\partial \mathscr{L}}{\partial(\partial_0\Phi)}$ is the conjugate momentum. | \begin{equation} \label{qftcomm} [\Phi(x), \pi(y)]=\Phi(x) \pi(y) - \pi(y) \Phi(x) = i \delta(x-y) \end{equation} where $\delta(x-y)$ is the Dirac delta distribution and $\pi(y) = \frac{\partial \mathscr{L}}{\partial(\partial_0\Phi)}$ is the conjugate momentum. | ||
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| \begin{equation}\label{KGsol} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right)\end{equation} | \begin{equation}\label{KGsol} \Phi(x)= \int \mathrm{d }k^3 \frac{1}{(2\pi)^3 2\omega_k} \left( a(k){\mathrm{e }}^{ -i(k x)} + a^\dagger(k) {\mathrm{e }}^{ i(kx)}\right)\end{equation} | ||
| - | Now, if we combine this solution with the canonical commutation relation we see that $a(k)$ and $a^\dagger(k)$ can‘t be numbers, but must be __operators__. Using $[\Phi(x), \pi(y)] = i \delta(x-y)$ we cam we can compute | + | Now, if we combine this solution with the canonical commutation relation we see that $a(k)$ and $a^\dagger(k)$ can‘t be numbers, but must be __operators__. Using $[\Phi(x), \pi(y)] = i \delta(x-y)$ we can compute |
| $$ [a^\dagger(k), a(k)] = i \delta(x-y) .$$ | $$ [a^\dagger(k), a(k)] = i \delta(x-y) .$$ | ||
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| So what are these operators? | So what are these operators? | ||
| - | To answer this question, we next look at something that understand: energy. | + | To answer this question, we next look at something that we understand: energy. |
| - | We can compute, using the Lagrangian, the corresponding Hamiltonian, which represents the energy and is the conserved quantity that follows from invariance under time-translation, if we use Noether‘s theorem. | + | Using the Lagrangian, we can compute the corresponding Hamiltonian, which represents the energy and is the conserved quantity that follows from invariance under time-translation, according to Noether‘s theorem. |
| For example, for spin $0$ fields we have\begin{equation} \label{hamil} H= \frac{1}{2} \Big( \big(\partial_0 \Phi \big)^2 + ( \partial_i \Phi )^2 + m \Phi^2 \Big). \end{equation}The fields are operators and therefore the Hamiltonian is an operator. For reasons explained above we call it the energy operator, which means if we act with the Hamiltonian on an abstract state $ | \Psi \rangle$ that describes the system in question, we get the energy of the system:$$ H | \Psi \rangle = E | \Psi \rangle $$ | For example, for spin $0$ fields we have\begin{equation} \label{hamil} H= \frac{1}{2} \Big( \big(\partial_0 \Phi \big)^2 + ( \partial_i \Phi )^2 + m \Phi^2 \Big). \end{equation}The fields are operators and therefore the Hamiltonian is an operator. For reasons explained above we call it the energy operator, which means if we act with the Hamiltonian on an abstract state $ | \Psi \rangle$ that describes the system in question, we get the energy of the system:$$ H | \Psi \rangle = E | \Psi \rangle $$ | ||
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| $$ H a^\dagger(k') a^\dagger(k') | 0\rangle = 2\omega_{k'} a^\dagger(k') a^\dagger(k') | 0\rangle $$ | $$ H a^\dagger(k') a^\dagger(k') | 0\rangle = 2\omega_{k'} a^\dagger(k') a^\dagger(k') | 0\rangle $$ | ||
| - | Recall that $a^\dagger(k') and $a(k') $ are the operator parts of our quantum fields. Here we learn that quantum fields create and annihilate particles! | + | Recall that $a^\dagger(k')$ and $a(k')$ are the operator parts of our quantum fields. Here we learn that quantum fields create and annihilate particles! |
| $| 0\rangle$ is an empty system | $| 0\rangle$ is an empty system | ||
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| As noted above, we can compute the Hamiltonian $H$ from the corresponding Lagrangian and the Lagrangian is what defines our physical theory. | As noted above, we can compute the Hamiltonian $H$ from the corresponding Lagrangian and the Lagrangian is what defines our physical theory. | ||
| - | The derivation of the Lagrangian that describe the interactions of particles is done using [[theories:gauge_theory|gauge theory]]. | + | The derivation of the Lagrangian that describe the interactions of particles is done using [[models:gauge_theory|gauge theory]]. |
| This yields one Lagrangian (and therefore one Hamiltonian) for electromagnetic interactions, one for weak and one for strong interactions. For example the Lagrangian, describing electromagnetic interactions is$$ H = \int d^3x \left( g A_\mu \bar \Psi \gamma^\mu \Psi \right) $$Here $A_\mu$ describes a spin $1$ field and $\Psi$ a spin $ \frac{1}{2}$ field. | This yields one Lagrangian (and therefore one Hamiltonian) for electromagnetic interactions, one for weak and one for strong interactions. For example the Lagrangian, describing electromagnetic interactions is$$ H = \int d^3x \left( g A_\mu \bar \Psi \gamma^\mu \Psi \right) $$Here $A_\mu$ describes a spin $1$ field and $\Psi$ a spin $ \frac{1}{2}$ field. | ||
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| - http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/253b_Lectures.pdf | - http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/253b_Lectures.pdf | ||
| - [[http://www.pp.rhul.ac.uk/~kauer/projects/scripts/ohl.pdf|Feynman Diagrams For Pedestrians]] by Thorsten Ohl | - [[http://www.pp.rhul.ac.uk/~kauer/projects/scripts/ohl.pdf|Feynman Diagrams For Pedestrians]] by Thorsten Ohl | ||
| + | - [[https://arxiv.org/abs/1110.5013|Notes from Sidney Coleman’s Physics 253a]] by Sidney Coleman | ||
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| * For a nice overview, see "[[http://indico.ipmu.jp/indico/event/134/contribution/17/material/slides/0.pdf|What is Quantum Field Theory]]", by Yuji Tachikawa | * For a nice overview, see "[[http://indico.ipmu.jp/indico/event/134/contribution/17/material/slides/0.pdf|What is Quantum Field Theory]]", by Yuji Tachikawa | ||
| - | * and in particular [[https://www.physicsforums.com/insights/causal-perturbation-theory/|How I Learned to Stop Worrying and Love QFT]] by Mario Flory. | + | * and in particular [[https://www.physicsforums.com/insights/causal-perturbation-theory/|How I Learned to Stop Worrying and Love QFT]] by Mario Flory |
| + | * and [[http://www.fuw.edu.pl/~kostecki/daniel_ranard_essay.pdf|An introduction to rigorous formulations of quantum field theory]] by Ranard | ||
| See also: | See also: | ||
| * [[https://www.physicsforums.com/insights/paqft-idea-references/|Introduction to Perturbative Quantum Field Theory]] by Urs Schreiber, | * [[https://www.physicsforums.com/insights/paqft-idea-references/|Introduction to Perturbative Quantum Field Theory]] by Urs Schreiber, | ||
| * [[https://www.physicsforums.com/insights/causal-perturbation-theory/|Causal Perturbation Theory]] by Arnold Neumeier | * [[https://www.physicsforums.com/insights/causal-perturbation-theory/|Causal Perturbation Theory]] by Arnold Neumeier | ||
| - | * [[http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf|THE CONCEPTUAL BASIS OFQUANTUM FIELD THEORY]] by Gerard ’t Hooft | + | * [[http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf|The Conceptual Basis of Quantum Field Theory]] by Gerard ’t Hooft |
| * “The Conceptual Framework of Quantum Field Theory” by A. Duncan | * “The Conceptual Framework of Quantum Field Theory” by A. Duncan | ||
| * See also [[http://chaosbook.org/FieldTheory/QMlectures/lectQM.pdf|QUANTUM FIELD THEORY a cyclist tour]] Predrag Cvitanovic | * See also [[http://chaosbook.org/FieldTheory/QMlectures/lectQM.pdf|QUANTUM FIELD THEORY a cyclist tour]] Predrag Cvitanovic | ||
| * [[ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/07/where_we_are_LNP.pdf|Quantum Field Theory: Where We Are]] by K. Fredenhagen | * [[ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/07/where_we_are_LNP.pdf|Quantum Field Theory: Where We Are]] by K. Fredenhagen | ||
| + | * [[https://arxiv.org/abs/1208.1428|Perturbative algebraic quantum field theory]] by K. Fredenhagen and K. Rejzner | ||
| + | * Quantum Mechanics and Quantum Field Theory: A Mathematical Primer by J. Dimock | ||
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| * The Quantum Theory of Fields, Volume 1-3 by Weinberg | * The Quantum Theory of Fields, Volume 1-3 by Weinberg | ||
| * Local Quantum Physics by Haag | * Local Quantum Physics by Haag | ||
| - | * Sidney Coleman, Aspects of Symmetry | + | * Aspects of Symmetry by Sidney Coleman |
| - | * Robin Ticciati, Quantum Field Theory for Mathematicians | + | * Quantum Field Theory for Mathematicians by Robin Ticciati |
| * The Global Approach to Quantum Field Theory by DeWitt | * The Global Approach to Quantum Field Theory by DeWitt | ||
| * Mathematical Aspects of Quantum Field Theory by Edson de Faria,Welington de Melo | * Mathematical Aspects of Quantum Field Theory by Edson de Faria,Welington de Melo | ||
| + | * Quantum Fields and Strings A Course for Mathematicians - edited by P. Deligne et. al. | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
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| <blockquote> | <blockquote> | ||
| "There are no particles, there are only fields" | "There are no particles, there are only fields" | ||
| - | |||
| <cite>https://arxiv.org/abs/1204.4616</cite> | <cite>https://arxiv.org/abs/1204.4616</cite> | ||
| </blockquote> | </blockquote> | ||
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| <blockquote> | <blockquote> | ||
| In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles. | In its mature form, the idea of quantum field theory is that quantum fields are the basic ingredients of the universe, and particles are just bundles of energy and momentum of the fields. In a relativistic theory the wave function is a functional of these fields, not a function of particle coordinates. Quantum field theory hence led to a more unified view of nature than the old dualistic interpretation in terms of both fields and particles. | ||
| - | |||
| <cite>[[https://arxiv.org/pdf/hep-th/9702027v1.pdf|What is Quantum Field Theory, and What Did We Think It Is?]] by S. Weinberg</cite> | <cite>[[https://arxiv.org/pdf/hep-th/9702027v1.pdf|What is Quantum Field Theory, and What Did We Think It Is?]] by S. Weinberg</cite> | ||
| </blockquote> | </blockquote> | ||
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| <blockquote> | <blockquote> | ||
| We have no better way of describing elementary particles than quantum field theory. A quantum field in general is an assembly of an infinite number of interacting harmonic oscillators. Excitations of such oscillators are associated with particles. The special importance of the harmonic oscillator follows from the fact that its excitation spectrum is additive, i.e. if $E_1$ and $E_2$ are energy levels above the ground state then $E_1 + E_2$ will be an energy level as well. It is precisely this property that we expect to be true for a system of elementary particles. | We have no better way of describing elementary particles than quantum field theory. A quantum field in general is an assembly of an infinite number of interacting harmonic oscillators. Excitations of such oscillators are associated with particles. The special importance of the harmonic oscillator follows from the fact that its excitation spectrum is additive, i.e. if $E_1$ and $E_2$ are energy levels above the ground state then $E_1 + E_2$ will be an energy level as well. It is precisely this property that we expect to be true for a system of elementary particles. | ||
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| <cite>A. M. Polyakov, “Gauge Fields and Strings”, 1987</cite> | <cite>A. M. Polyakov, “Gauge Fields and Strings”, 1987</cite> | ||
| </blockquote> | </blockquote> | ||
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| <blockquote> | <blockquote> | ||
| Undoubtedly the single most profound fact about Nature that quantum field theory uniquely explains is the existence of different, yet indistinguishable, copies of elementary particles. Two electrons anywhere in the Universe, whatever their origin or history, are observed to have exactly the same properties. We understand this as a consequence of the fact that both are excitations of the same underlying ur-stuff, the electron field. | Undoubtedly the single most profound fact about Nature that quantum field theory uniquely explains is the existence of different, yet indistinguishable, copies of elementary particles. Two electrons anywhere in the Universe, whatever their origin or history, are observed to have exactly the same properties. We understand this as a consequence of the fact that both are excitations of the same underlying ur-stuff, the electron field. | ||
| - | |||
| <cite>[[https://arxiv.org/abs/hep-th/9803075|Quantum Field Theory]] by Frank Wilczek</cite> | <cite>[[https://arxiv.org/abs/hep-th/9803075|Quantum Field Theory]] by Frank Wilczek</cite> | ||
| </blockquote> | </blockquote> | ||
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| <tabbox History> | <tabbox History> | ||
| * See: [[https://arxiv.org/pdf/1503.05007.pdf|The Evolution of Quantum Field Theory]] by Gerard ’t Hooft | * See: [[https://arxiv.org/pdf/1503.05007.pdf|The Evolution of Quantum Field Theory]] by Gerard ’t Hooft | ||
| + | * https://plus.maths.org/content/brief-history-quantum-field-theory | ||
| </tabbox> | </tabbox> | ||
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