In particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the unification energy, on the order of 100 GeV, they would merge into a single electroweak force.
In particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons.
The simplest description of the mechanism adds a Higgs field to the Standard Model gauge theory. The symmetry breaking triggers conversion of the longitudinal field component to the Higgs boson, which interacts with itself and (at least a part of) the other fields in the theory, so as to produce mass terms for the Z and W bosons.
A colleague of yours has given you mathematical expressions for the f.pdfarjuntiwari586
A colleague of yours has given you mathematical expressions for the following electromagnetic
fields that they have measured. i. E(x, y, z, t)z squareroot x+ u_t sinc (x + u-p t) U (x + u_pt),
where u_p = 1/squareroot epsilon_0 mu_0, and U (pi) is the unit step function. ii. H(x, y, z, t) =
zH_ye -(z -ct/2) cos (z - ct/2), where c = 3 times 10^8 m/sec iii. E (x, y, z, t) = x cos (pi y) cos
(z^2 - u_p t) v. E (r, theta, z, t) = E_0 cos (omega t - beta r) In all of these cases, analyze and
state each of the following properties of these EM fields; (a) Which of the fields are plane
waves? In each case, give a detailed explanation of your reasoning and state which plane (i.e. xz-
plane, ... etc.) the waves are measured. (b) Which of the plane waves are uniform plane waves?
Why? (c) Calculate what the velocity of each of the uniform plane waves is, which direction it is
propagating in, and which waves are travelling in free space? (d) List the plane waves (either
uniform or non-uniform) that are time harmonic and explain why? (e) In the case of the uniform
plane waves, state (write down in differential form) which one of Maxwell\'s equations and
whose law it is to find the corresponding magnetic or electric field. Calculate the magnetic or
electric field intensity from this law. Assume that the uniform plane waves are propagating in a
linear, homogeneous, isotropic (LHI) media that is lossless.
Solution
The Relation Between Expressions for Time-Dependent Electromagnetic Fields Given by
Jefimenko and by Panofsky and Phillips Kirk T. McDonald Joseph Henry Laboratories,
Princeton University, Princeton, NJ 08544 (Dec. 5, 1996; updated May 7, 2016) Abstract The
expressions of Jefimenko for the electromagnetic fields E and B in terms of source charge and
current densities and J, which have received much recent attention in the American Journal of
Physics, appeared previously in sec. 14.3 of the book Classical Electricity and Magnetism by
Panofsky and Phillips. The latter develop these expressions further into a form that gives greater
emphasis to the radiation fields. This Note presents a derivation of the various expressions and
discusses an apparent paradox in applying Panofsky and Phillips’ result to static situations. 1
Introduction A general method of calculation of time-dependent electromagnetic fields was
given by Lorenz in 1867 [1], in which the retarded potentials were first introduced.1 These are
(x, t) = [(x , t )] R d3x , and A(x, t) = 1 c [J(x , t )] R d3x , (1) where and A are the scalar and
vector potentials in Gaussian units ,2 and J are the charge and current densities, R = |R| with R =
xx , and a pair of brackets, [ ], implies the quantity within is to be evaluated at the retarded time t
= t R/c with c being the speed of light in vacuum. Lorenz did not explicitly display the electric
field E and the magnetic field B, although he noted they could be obtained via E = 1 c A t , and
B = × A. (3) Had Lorenz’ work been better received by Max.
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
Em Ciência da Computação, uma função de mão única ou função de sentido único é uma função que é fácil de calcular para qualquer entrada (qualquer valor do seu domínio), mas difícil de inverter dada a imagem de uma entrada aleatória. Aqui "fácil" e "difícil" são entendidos em termos da teoria da complexidade computacional, especificamente a teoria dos problemas de tempo polinomial. Não sendo um-para-um não é considerado suficiente para um função ser chamada de mão única.
Galalactic Dynamics- Orbits in a Logarithmic Dark Matter Halo PotentialPritam Kalbhor
This is the slide show for study of orbits in a Logarithmic dark matter halo potential. I have founded orbits in a both static and rotatiting logarithmic potential
In particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different aspects of the same force. Above the unification energy, on the order of 100 GeV, they would merge into a single electroweak force.
In particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons.
The simplest description of the mechanism adds a Higgs field to the Standard Model gauge theory. The symmetry breaking triggers conversion of the longitudinal field component to the Higgs boson, which interacts with itself and (at least a part of) the other fields in the theory, so as to produce mass terms for the Z and W bosons.
A colleague of yours has given you mathematical expressions for the f.pdfarjuntiwari586
A colleague of yours has given you mathematical expressions for the following electromagnetic
fields that they have measured. i. E(x, y, z, t)z squareroot x+ u_t sinc (x + u-p t) U (x + u_pt),
where u_p = 1/squareroot epsilon_0 mu_0, and U (pi) is the unit step function. ii. H(x, y, z, t) =
zH_ye -(z -ct/2) cos (z - ct/2), where c = 3 times 10^8 m/sec iii. E (x, y, z, t) = x cos (pi y) cos
(z^2 - u_p t) v. E (r, theta, z, t) = E_0 cos (omega t - beta r) In all of these cases, analyze and
state each of the following properties of these EM fields; (a) Which of the fields are plane
waves? In each case, give a detailed explanation of your reasoning and state which plane (i.e. xz-
plane, ... etc.) the waves are measured. (b) Which of the plane waves are uniform plane waves?
Why? (c) Calculate what the velocity of each of the uniform plane waves is, which direction it is
propagating in, and which waves are travelling in free space? (d) List the plane waves (either
uniform or non-uniform) that are time harmonic and explain why? (e) In the case of the uniform
plane waves, state (write down in differential form) which one of Maxwell\'s equations and
whose law it is to find the corresponding magnetic or electric field. Calculate the magnetic or
electric field intensity from this law. Assume that the uniform plane waves are propagating in a
linear, homogeneous, isotropic (LHI) media that is lossless.
Solution
The Relation Between Expressions for Time-Dependent Electromagnetic Fields Given by
Jefimenko and by Panofsky and Phillips Kirk T. McDonald Joseph Henry Laboratories,
Princeton University, Princeton, NJ 08544 (Dec. 5, 1996; updated May 7, 2016) Abstract The
expressions of Jefimenko for the electromagnetic fields E and B in terms of source charge and
current densities and J, which have received much recent attention in the American Journal of
Physics, appeared previously in sec. 14.3 of the book Classical Electricity and Magnetism by
Panofsky and Phillips. The latter develop these expressions further into a form that gives greater
emphasis to the radiation fields. This Note presents a derivation of the various expressions and
discusses an apparent paradox in applying Panofsky and Phillips’ result to static situations. 1
Introduction A general method of calculation of time-dependent electromagnetic fields was
given by Lorenz in 1867 [1], in which the retarded potentials were first introduced.1 These are
(x, t) = [(x , t )] R d3x , and A(x, t) = 1 c [J(x , t )] R d3x , (1) where and A are the scalar and
vector potentials in Gaussian units ,2 and J are the charge and current densities, R = |R| with R =
xx , and a pair of brackets, [ ], implies the quantity within is to be evaluated at the retarded time t
= t R/c with c being the speed of light in vacuum. Lorenz did not explicitly display the electric
field E and the magnetic field B, although he noted they could be obtained via E = 1 c A t , and
B = × A. (3) Had Lorenz’ work been better received by Max.
In this second lecture, I will discuss how to calculate polarization in terms of Berry phase, how to include GW correction in the real-time dynamics and electron-hole interaction.
Em Ciência da Computação, uma função de mão única ou função de sentido único é uma função que é fácil de calcular para qualquer entrada (qualquer valor do seu domínio), mas difícil de inverter dada a imagem de uma entrada aleatória. Aqui "fácil" e "difícil" são entendidos em termos da teoria da complexidade computacional, especificamente a teoria dos problemas de tempo polinomial. Não sendo um-para-um não é considerado suficiente para um função ser chamada de mão única.
Galalactic Dynamics- Orbits in a Logarithmic Dark Matter Halo PotentialPritam Kalbhor
This is the slide show for study of orbits in a Logarithmic dark matter halo potential. I have founded orbits in a both static and rotatiting logarithmic potential
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Causality from outside Time
Alfred Driessen
Talk presented at the 21th International Interdisciplinary Seminar,
Science and Society: Defining what is human
Netherhall House, London, 5-1-2019
Content
Introduction
Time in Relativity
Time in Quantum Mechanics
Conclusions
Conclusions from this study:
There are causes beyond the realm of science,
- they are not observable by physical or scientific means
- the effects of these causes, however, are observable by physical and scientific means.
Physics is not complete.
Causality from outside Time
Alfred Driessen
Talk presented at the 21th International Interdisciplinary Seminar, Science and Society: Defining what is human
Netherhall House, London, 5-1-2019
Content
Introduction
Time in Relativity
Time in Quantum Mechanics
Conclusions
Conclusions from this study:
There are causes beyond the realm of science,
- they are not observable by physical or scientific means
- the effects of these causes, however, are observable by physical and scientific means.
Physics is not complete.
Many computer graphics and Image Processing effects owe much of their realism to the study of fractals and noise. This short tutorial is based on over a decade of teaching and research interests, and will take a journey from the motion of a microscopic particle to the creation of imaginary planets.
Further resources at:
http://wiki.rcs.manchester.ac.uk/community/Fractal_Resources_Tutorial
Broken Time-Reversal Symmetry and Topological Order in Triplet SuperconductorsJorge Quintanilla
Jorge Quintanilla, "Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors" - Research seminar, Max Planck Institute for the Physics of Complex Systems (Dresden), 27 November 2014
Abstract:
The concept of broken symmetry is one of the cornerstones of modern physics, for which
superconductors stand out as a major paradigm. In conventional superconductors electrons form
isotropic singlet pairs that then condense into a coherent state, similar to that of photons in a laser.
We understand this in terms of the breaking of global gauge symmetry, which is the invariance of a
system under changes to the overall phase of its wave function. In unconventional superconductors,
however, more complex forms of pairing are possible, leading to additional broken symmetries and
even to topological forms of order that fall outside the broken-symmetry paradigm.
In this talk I will discuss such phenomena, making emphasis on triplet pairing and the spontaneous
breaking of time-reversal symmetry in some superconductors. I will pay particular attention to
large-facility experiments using muons to detect tiny magnetic fields inside superconducting
samples and group-theoretical arguments that enable us to constrain the type of pairing present in
the light of such experiments. I will also address the possibility of mixed singlet-triplet pairing
without broken time-reversal symmetry in superconductors whose crystal lattices lack a centre of
inversion, and predict bulk experimental signatures of topological transitions expected to occur in
such systems.
Talk given at Kobayashi-Maskawa Institute, Nagoya University, Japan.Peter Coles
Cosmic Anomalies
Observational measurements of the temperature variation of the Cosmic Microwave Background across the celestial sphere made by the Wilkinson Microwave Anisotropy Probe (WMAP) and, more recently Planck, have played a major part in establishing the standard "concordance" cosmological model. However, extensive statistical analysis of these data have also revealed some tantalising anomalies whose interpretation within the standard framework is by no means clear. In this talk, I'll discuss the significance of the evidence for some aspects of this anomalous behaviour, offer some possible theoretical models, and suggest how future measurements may provide firmer conclusions.
Searches for new physics at LHC within the Higgs sector. Step 2: Defining the...Raquel Gomez Ambrosio
We discuss the Effective field theory bottom-up approach, and show some examples of its application for VH production at LHC. We find some interesting results regarding the applicability of the perturbative expansion. Finally we discuss the Pseudo Observable approach as a tool for New Physics searches at LHC.
1.1 PRINCIPLE OF LEAST ACTION 640-213 MelatosChapter 1.docxpaynetawnya
1.1 PRINCIPLE OF LEAST ACTION 640-213 Melatos
Chapter 1
A New Perspective on F = ma
Reference: Feynman, Lectures on Physics, Volume 2, Chapter 19
1.1 Principle of Least Action
Throw a tennis ball (mass m) straight upwards. How does it “know” where to move?
Traditional perspective (local):
at every instant t, the ball accelerates in response to the net force F(t) acting at
time t only, not at some earlier or later time; we update its velocity according
to v(t + Δt) = v(t) + a(t)Δt and its position according to x(t + Δt) =
x(t) + v(t)Δt + a(t)(Δt)2/2, with a(t) = F(t)/m.
New perspective (global):
“try” lots of different paths and choose the one which extremises some “virtue”
What is this “virtue”? It is different for different natural laws. For example:
• propagation of light: minimise (usually) the travel time between two points
• Maxwell’s laws: minimise the difference between the electric and magnetic energies
• Einstein’s theory of gravity: minimise a messy function of the space-time curvature
1
1.1 PRINCIPLE OF LEAST ACTION 640-213 Melatos
For our tennis ball:
compute the kinetic energy at each point on the path, subtract the potential
energy at each point, and integrate over time along the whole path; the result
is smallest for the true path, smaller than for any other trial path between
the same two endpoints
S =
∫ t2
t1
dt
[
1
2
m
∣∣∣∣dx(t)dt
∣∣∣∣
2
− mgx(t)
]
(1.1)
S is called the action. (Units: J s.)
Exercise. What other famous physical constant has units of J s?
The integrand [...] in (1.1) is called the Lagrangian L. The law is called the Principle
of Least Action (POLA). Because the potential energy increases with altitude, the ball
“wants” to reach a high altitude as quickly as possible, in order to minimise S in the fixed
time available (t2 − t1). But if the ball goes too fast (which it must do if it is to rise very
high in the fixed time available), then the kinetic energy gets too big and outweighs the
benefits of the potential energy.
The true path is the best compromise which minimises S.
Exercise. Explain why a free particle (i.e., zero potential energy) travels at
uniform speed.
If this new perspective is to prove useful, it should provide a neat way to solve mathe-
matically for the path x(t), as a substitute for integrating F = mẍ directly. And it does!
We will see, in the first half of the course, how certain quantities derived easily from L
2
1.2 OPTIMAL PATH BY THE CALCULUS OF VARIATIONS 640-213 Melatos
are conserved, greatly simplifying the task of solving for x(t). The conserved quantities
correspond to symmetries in the physics of the problem, e.g., rotational symmetry ⇔
conservation of angular momentum. Sometimes the symmetries are “hidden” and would
have been hard to guess without the new, Lagrangian perspective. In these situations,
the beauty and utility of Lagrangian mechanics are simultaneously on full display.
1.2 Optimal path by the calculus of variations
Before learni ...
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
Causality from outside Time
Alfred Driessen
Talk presented at the 21th International Interdisciplinary Seminar,
Science and Society: Defining what is human
Netherhall House, London, 5-1-2019
Content
Introduction
Time in Relativity
Time in Quantum Mechanics
Conclusions
Conclusions from this study:
There are causes beyond the realm of science,
- they are not observable by physical or scientific means
- the effects of these causes, however, are observable by physical and scientific means.
Physics is not complete.
Causality from outside Time
Alfred Driessen
Talk presented at the 21th International Interdisciplinary Seminar, Science and Society: Defining what is human
Netherhall House, London, 5-1-2019
Content
Introduction
Time in Relativity
Time in Quantum Mechanics
Conclusions
Conclusions from this study:
There are causes beyond the realm of science,
- they are not observable by physical or scientific means
- the effects of these causes, however, are observable by physical and scientific means.
Physics is not complete.
Many computer graphics and Image Processing effects owe much of their realism to the study of fractals and noise. This short tutorial is based on over a decade of teaching and research interests, and will take a journey from the motion of a microscopic particle to the creation of imaginary planets.
Further resources at:
http://wiki.rcs.manchester.ac.uk/community/Fractal_Resources_Tutorial
Broken Time-Reversal Symmetry and Topological Order in Triplet SuperconductorsJorge Quintanilla
Jorge Quintanilla, "Broken Time-Reversal Symmetry and Topological Order in Triplet Superconductors" - Research seminar, Max Planck Institute for the Physics of Complex Systems (Dresden), 27 November 2014
Abstract:
The concept of broken symmetry is one of the cornerstones of modern physics, for which
superconductors stand out as a major paradigm. In conventional superconductors electrons form
isotropic singlet pairs that then condense into a coherent state, similar to that of photons in a laser.
We understand this in terms of the breaking of global gauge symmetry, which is the invariance of a
system under changes to the overall phase of its wave function. In unconventional superconductors,
however, more complex forms of pairing are possible, leading to additional broken symmetries and
even to topological forms of order that fall outside the broken-symmetry paradigm.
In this talk I will discuss such phenomena, making emphasis on triplet pairing and the spontaneous
breaking of time-reversal symmetry in some superconductors. I will pay particular attention to
large-facility experiments using muons to detect tiny magnetic fields inside superconducting
samples and group-theoretical arguments that enable us to constrain the type of pairing present in
the light of such experiments. I will also address the possibility of mixed singlet-triplet pairing
without broken time-reversal symmetry in superconductors whose crystal lattices lack a centre of
inversion, and predict bulk experimental signatures of topological transitions expected to occur in
such systems.
Talk given at Kobayashi-Maskawa Institute, Nagoya University, Japan.Peter Coles
Cosmic Anomalies
Observational measurements of the temperature variation of the Cosmic Microwave Background across the celestial sphere made by the Wilkinson Microwave Anisotropy Probe (WMAP) and, more recently Planck, have played a major part in establishing the standard "concordance" cosmological model. However, extensive statistical analysis of these data have also revealed some tantalising anomalies whose interpretation within the standard framework is by no means clear. In this talk, I'll discuss the significance of the evidence for some aspects of this anomalous behaviour, offer some possible theoretical models, and suggest how future measurements may provide firmer conclusions.
Searches for new physics at LHC within the Higgs sector. Step 2: Defining the...Raquel Gomez Ambrosio
We discuss the Effective field theory bottom-up approach, and show some examples of its application for VH production at LHC. We find some interesting results regarding the applicability of the perturbative expansion. Finally we discuss the Pseudo Observable approach as a tool for New Physics searches at LHC.
1.1 PRINCIPLE OF LEAST ACTION 640-213 MelatosChapter 1.docxpaynetawnya
1.1 PRINCIPLE OF LEAST ACTION 640-213 Melatos
Chapter 1
A New Perspective on F = ma
Reference: Feynman, Lectures on Physics, Volume 2, Chapter 19
1.1 Principle of Least Action
Throw a tennis ball (mass m) straight upwards. How does it “know” where to move?
Traditional perspective (local):
at every instant t, the ball accelerates in response to the net force F(t) acting at
time t only, not at some earlier or later time; we update its velocity according
to v(t + Δt) = v(t) + a(t)Δt and its position according to x(t + Δt) =
x(t) + v(t)Δt + a(t)(Δt)2/2, with a(t) = F(t)/m.
New perspective (global):
“try” lots of different paths and choose the one which extremises some “virtue”
What is this “virtue”? It is different for different natural laws. For example:
• propagation of light: minimise (usually) the travel time between two points
• Maxwell’s laws: minimise the difference between the electric and magnetic energies
• Einstein’s theory of gravity: minimise a messy function of the space-time curvature
1
1.1 PRINCIPLE OF LEAST ACTION 640-213 Melatos
For our tennis ball:
compute the kinetic energy at each point on the path, subtract the potential
energy at each point, and integrate over time along the whole path; the result
is smallest for the true path, smaller than for any other trial path between
the same two endpoints
S =
∫ t2
t1
dt
[
1
2
m
∣∣∣∣dx(t)dt
∣∣∣∣
2
− mgx(t)
]
(1.1)
S is called the action. (Units: J s.)
Exercise. What other famous physical constant has units of J s?
The integrand [...] in (1.1) is called the Lagrangian L. The law is called the Principle
of Least Action (POLA). Because the potential energy increases with altitude, the ball
“wants” to reach a high altitude as quickly as possible, in order to minimise S in the fixed
time available (t2 − t1). But if the ball goes too fast (which it must do if it is to rise very
high in the fixed time available), then the kinetic energy gets too big and outweighs the
benefits of the potential energy.
The true path is the best compromise which minimises S.
Exercise. Explain why a free particle (i.e., zero potential energy) travels at
uniform speed.
If this new perspective is to prove useful, it should provide a neat way to solve mathe-
matically for the path x(t), as a substitute for integrating F = mẍ directly. And it does!
We will see, in the first half of the course, how certain quantities derived easily from L
2
1.2 OPTIMAL PATH BY THE CALCULUS OF VARIATIONS 640-213 Melatos
are conserved, greatly simplifying the task of solving for x(t). The conserved quantities
correspond to symmetries in the physics of the problem, e.g., rotational symmetry ⇔
conservation of angular momentum. Sometimes the symmetries are “hidden” and would
have been hard to guess without the new, Lagrangian perspective. In these situations,
the beauty and utility of Lagrangian mechanics are simultaneously on full display.
1.2 Optimal path by the calculus of variations
Before learni ...
(If visualization is slow, please try downloading the file.)
Part 2 of a tutorial given in the Brazilian Physical Society meeting, ENFMC. Abstract: Density-functional theory (DFT) was developed 50 years ago, connecting fundamental quantum methods from early days of quantum mechanics to our days of computer-powered science. Today DFT is the most widely used method in electronic structure calculations. It helps moving forward materials sciences from a single atom to nanoclusters and biomolecules, connecting solid-state, quantum chemistry, atomic and molecular physics, biophysics and beyond. In this tutorial, I will try to clarify this pathway under a historical view, presenting the DFT pillars and its building blocks, namely, the Hohenberg-Kohn theorem, the Kohn-Sham scheme, the local density approximation (LDA) and generalized gradient approximation (GGA). I would like to open the black box misconception of the method, and present a more pedagogical and solid perspective on DFT.
1. The Origin of Complex Structures in
QFT on Curved Space-Times
Philip Tillman
Tuesday, January 8, 2008
P. Tillman () The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 1 / 16
2. Outline
1 Technical Di¢ culties
2 Complex Structures and Time Evolution
3 Construction of the Fock Space
4 The Unruh E¤ect
5 The Quantum Field and Particles
6 Comments/Open Questions
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16
3. Outline
1 Technical Di¢ culties
2 Complex Structures and Time Evolution
3 Construction of the Fock Space
4 The Unruh E¤ect
5 The Quantum Field and Particles
6 Comments/Open Questions
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16
4. Outline
1 Technical Di¢ culties
2 Complex Structures and Time Evolution
3 Construction of the Fock Space
4 The Unruh E¤ect
5 The Quantum Field and Particles
6 Comments/Open Questions
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16
5. Outline
1 Technical Di¢ culties
2 Complex Structures and Time Evolution
3 Construction of the Fock Space
4 The Unruh E¤ect
5 The Quantum Field and Particles
6 Comments/Open Questions
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16
6. Outline
1 Technical Di¢ culties
2 Complex Structures and Time Evolution
3 Construction of the Fock Space
4 The Unruh E¤ect
5 The Quantum Field and Particles
6 Comments/Open Questions
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16
7. Outline
1 Technical Di¢ culties
2 Complex Structures and Time Evolution
3 Construction of the Fock Space
4 The Unruh E¤ect
5 The Quantum Field and Particles
6 Comments/Open Questions
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 2 / 16
8. Technical Di¢ culties (1)
1 No Poincare Symmetry
INo "preferred" time and No global Fourier
transforms
2 Unboundedness of operators:
[ˆx, ˆp] = i h
m
ei ˆx
ei ˆp
= e [ˆx,ˆp]/2
eˆx+ˆp
| {z }
Weyl Relations
+
strong
continuity
3 Operator-valued distributions ˆφ (x) are singular
ISmear them out by test functions f in ˆφ (f )
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 3 / 16
9. Technical Di¢ culties (1)
1 No Poincare Symmetry
INo "preferred" time and No global Fourier
transforms
2 Unboundedness of operators:
[ˆx, ˆp] = i h
m
ei ˆx
ei ˆp
= e [ˆx,ˆp]/2
eˆx+ˆp
| {z }
Weyl Relations
+
strong
continuity
3 Operator-valued distributions ˆφ (x) are singular
ISmear them out by test functions f in ˆφ (f )
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 3 / 16
10. Technical Di¢ culties (1)
1 No Poincare Symmetry
INo "preferred" time and No global Fourier
transforms
2 Unboundedness of operators:
[ˆx, ˆp] = i h
m
ei ˆx
ei ˆp
= e [ˆx,ˆp]/2
eˆx+ˆp
| {z }
Weyl Relations
+
strong
continuity
3 Operator-valued distributions ˆφ (x) are singular
ISmear them out by test functions f in ˆφ (f )
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 3 / 16
11. Technical Di¢ culties (2)
4. Well-posed initial value formulation
ISolutions of the …eld equations are well-de…ned
everywhere in space-time
IDe…ne space-time to be globally hyperbolic,
M = Σ R
5. Unitary Inequivalent constructions of Hilbert spaces
6. Renormalization
7. Relearn or entirely unlearn basic notions of QFT
IE.g.) The "particle notion" must be unlearned
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 4 / 16
12. Technical Di¢ culties (2)
4. Well-posed initial value formulation
ISolutions of the …eld equations are well-de…ned
everywhere in space-time
IDe…ne space-time to be globally hyperbolic,
M = Σ R
5. Unitary Inequivalent constructions of Hilbert spaces
6. Renormalization
7. Relearn or entirely unlearn basic notions of QFT
IE.g.) The "particle notion" must be unlearned
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 4 / 16
13. Technical Di¢ culties (2)
4. Well-posed initial value formulation
ISolutions of the …eld equations are well-de…ned
everywhere in space-time
IDe…ne space-time to be globally hyperbolic,
M = Σ R
5. Unitary Inequivalent constructions of Hilbert spaces
6. Renormalization
7. Relearn or entirely unlearn basic notions of QFT
IE.g.) The "particle notion" must be unlearned
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 4 / 16
14. Technical Di¢ culties (2)
4. Well-posed initial value formulation
ISolutions of the …eld equations are well-de…ned
everywhere in space-time
IDe…ne space-time to be globally hyperbolic,
M = Σ R
5. Unitary Inequivalent constructions of Hilbert spaces
6. Renormalization
7. Relearn or entirely unlearn basic notions of QFT
IE.g.) The "particle notion" must be unlearned
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 4 / 16
15. Complex Structures and Time Evolution
The Real Scalar (KG) Field:
rµrµ
+ m2
φ = 0
The Schrödinger equation for a single particle (φ+) in a
curved space-time is the de…nition of ˆH:
ˆHφ+
def
= iLX φ+
LX is the Lie derivative in the Xa
(time) direction, h = 1
The complex structure (Ashtekar-Magnon):
J
def
= LX / ( LX LX )1/2
, J2
= 1
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 5 / 16
16. Complex Structures and Time Evolution
Splitting a solution φ into positive/negative frequency
solutions:
φ = φ+ + φ
Jφ = iφ
Positive frequencies are particles states
Negative frequencies are anti-particles states
E.g.) Minkowski Space LX = ∂/∂τ
Jφ = J u (x) e iωτ
+ u (x) e+iωτ
= (+i) u (x) e iωτ
+ ( i) u (x) eiωτ
The decomposition φ = φ+ + φ and is related to the
choice of J.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 6 / 16
17. Construction of the Fock Space
Given a single particle Hilbert space H then de…ne the
Fock space F:
F
def
= ?|{z}
vacuum
H|{z}
single particle
H H| {z }
two particle
The construction of H is not complete. . .
We need an inner-product h, iH for expectation values.
F The choice of H is also related to choice of J because
H is the space of all φ+ and Jφ+ = +iφ+.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 7 / 16
18. The Symplectic Structure
To construct an inner product we use the symplectic
structure Ω ( , ).
Let S be the set of all fΦ = (φ, π)g that satisfy the
…eld equations (the space of solutions).
Fact due the form of the KG …eld equation:
S is a symplectic vector space.
Ω (Φ1, Φ2) =
Z
Σ0
(π1φ2 π2φ1) d3
x
=
Z
Σ0
((raφ1) φ2 (raφ2) φ1) na
p
hd3
x
h is the spacial metric on the Cauchy surface Σ0.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 8 / 16
19. The Inner Product
The single particle Hilbert space H inner product:
2 hΦ1, Φ2iH
def
= Ω (JΦ1, Φ2) iΩ (Φ1, Φ2)
H is de…ned by the Cauchy completion Sµ of S in the
positive de…nite norm:
µ (Φ1, Φ2)
def
= Ω (JΦ1, Φ2)
We now can de…ne the Fock space F:
F
def
= ?|{z}
vacuum
H|{z}
single particle
H H| {z }
two particle
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times 01/08/08 9 / 16
20. What We Have Learned So Far
1 The choice of time evolution (Lie derivative in
X-direction) is directly related to the choice of
complex structure:
J
def
= LX / ( LX LX )1/2
, J2
= 1
2 Also the choice of J de…nes our single particle
Hilbert space (hence our Fock space):
H
def
= φ+j φ+ 2 Sµ, Jφ+ = +iφ+ , h, iH : H ! C
where Sµ is the closure of S in µ ( , ) = Ω (J , ).
3 Next, an interesting physical consequence. . .
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 10 / 16
21. What We Have Learned So Far
1 The choice of time evolution (Lie derivative in
X-direction) is directly related to the choice of
complex structure:
J
def
= LX / ( LX LX )1/2
, J2
= 1
2 Also the choice of J de…nes our single particle
Hilbert space (hence our Fock space):
H
def
= φ+j φ+ 2 Sµ, Jφ+ = +iφ+ , h, iH : H ! C
where Sµ is the closure of S in µ ( , ) = Ω (J , ).
3 Next, an interesting physical consequence. . .
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 10 / 16
22. What We Have Learned So Far
1 The choice of time evolution (Lie derivative in
X-direction) is directly related to the choice of
complex structure:
J
def
= LX / ( LX LX )1/2
, J2
= 1
2 Also the choice of J de…nes our single particle
Hilbert space (hence our Fock space):
H
def
= φ+j φ+ 2 Sµ, Jφ+ = +iφ+ , h, iH : H ! C
where Sµ is the closure of S in µ ( , ) = Ω (J , ).
3 Next, an interesting physical consequence. . .
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 10 / 16
23. The Unruh E¤ect
In Minkowski space take 2 observers with particle
detectors. An inertial observer, and a uniformly
accelerating (Rindler) observer:
PICTURE
The inertial observer measures the state of the universe
on Σ to be the (Minkowski) vacuum j?Mi at point p.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 11 / 16
24. The Unruh E¤ect
The Rindler observer de…nes positive frequency particles
di¤erently:
φ = φ
(R)
+ + φ
(R)
as opposed to:
φ = φ
(M)
+ + φ
(M)
Two quantizations Q1 and Q2:
φ
Q1
! ˆφ1 = ˆφ
(M)
1+ + ˆφ
(M)
1
Q2
& ˆφ2 = ˆφ
(R)
2+ + ˆφ
(R)
2
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 12 / 16
25. The Unruh E¤ect
Bogoliubov transformation between the two:
U ˆφ1U 1
= ˆφ2
Consequently in the state j?Mi, the Rindler observer
measures Rindler particles φ
(R)
+ :
U j?Mi = c j?R i j2 particlesi j4 particlesi
Conundrum: The Rindler observer will measure
particles (ˆφ
(R)
2+ ) when the inertial observer would
measure none (vacuum j?Mi)
Explanation: Each observation of a Rindler
particle, is "seen" as an emission of a Minkowski
particle by the inertial observer (Unruh and Wald).
Weird!
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 13 / 16
26. The Unruh E¤ect
Bogoliubov transformation between the two:
U ˆφ1U 1
= ˆφ2
Consequently in the state j?Mi, the Rindler observer
measures Rindler particles φ
(R)
+ :
U j?Mi = c j?R i j2 particlesi j4 particlesi
Conundrum: The Rindler observer will measure
particles (ˆφ
(R)
2+ ) when the inertial observer would
measure none (vacuum j?Mi)
Explanation: Each observation of a Rindler
particle, is "seen" as an emission of a Minkowski
particle by the inertial observer (Unruh and Wald).
Weird!
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 13 / 16
27. The Unruh E¤ect
Bogoliubov transformation between the two:
U ˆφ1U 1
= ˆφ2
Consequently in the state j?Mi, the Rindler observer
measures Rindler particles φ
(R)
+ :
U j?Mi = c j?R i j2 particlesi j4 particlesi
Conundrum: The Rindler observer will measure
particles (ˆφ
(R)
2+ ) when the inertial observer would
measure none (vacuum j?Mi)
Explanation: Each observation of a Rindler
particle, is "seen" as an emission of a Minkowski
particle by the inertial observer (Unruh and Wald).
Weird!
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 13 / 16
28. The Unruh E¤ect
Why isn’t U j?Mi = j?R i?
Because positive frequency modes mix with negative
ones Bk 6= 0:
φ
(M)
+ = ∑
k
Ak φ
(R)
+k + Bk φ
(R)
k
Why aren’t all de…nitions of vacuums the
equivalent?
Ashtekar-Magnon: Ask the opposite question.
Why is the de…nition the same for all inertial
observers?
A: Because of a time-like Killing symmetry
) positive and negative modes don’t mix
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 14 / 16
29. The Unruh E¤ect
Why isn’t U j?Mi = j?R i?
Because positive frequency modes mix with negative
ones Bk 6= 0:
φ
(M)
+ = ∑
k
Ak φ
(R)
+k + Bk φ
(R)
k
Why aren’t all de…nitions of vacuums the
equivalent?
Ashtekar-Magnon: Ask the opposite question.
Why is the de…nition the same for all inertial
observers?
A: Because of a time-like Killing symmetry
) positive and negative modes don’t mix
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 14 / 16
30. The Unruh E¤ect
Why isn’t U j?Mi = j?R i?
Because positive frequency modes mix with negative
ones Bk 6= 0:
φ
(M)
+ = ∑
k
Ak φ
(R)
+k + Bk φ
(R)
k
Why aren’t all de…nitions of vacuums the
equivalent?
Ashtekar-Magnon: Ask the opposite question.
Why is the de…nition the same for all inertial
observers?
A: Because of a time-like Killing symmetry
) positive and negative modes don’t mix
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 14 / 16
31. The Unruh E¤ect
Why isn’t U j?Mi = j?R i?
Because positive frequency modes mix with negative
ones Bk 6= 0:
φ
(M)
+ = ∑
k
Ak φ
(R)
+k + Bk φ
(R)
k
Why aren’t all de…nitions of vacuums the
equivalent?
Ashtekar-Magnon: Ask the opposite question.
Why is the de…nition the same for all inertial
observers?
A: Because of a time-like Killing symmetry
) positive and negative modes don’t mix
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 14 / 16
32. The Unruh E¤ect
Why isn’t U j?Mi = j?R i?
Because positive frequency modes mix with negative
ones Bk 6= 0:
φ
(M)
+ = ∑
k
Ak φ
(R)
+k + Bk φ
(R)
k
Why aren’t all de…nitions of vacuums the
equivalent?
Ashtekar-Magnon: Ask the opposite question.
Why is the de…nition the same for all inertial
observers?
A: Because of a time-like Killing symmetry
) positive and negative modes don’t mix
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 14 / 16
33. The Quantum Field and Particles
The Relativity Principle: All observers are equivalent,
including non-inertial and inertial.
Consequences:
No "universal" de…nition of particles and No
"universal" de…nition of vacuum
Need observer dependent de…nition of particles and
vacuums state.
The …eld is more fundamental than particles:
The "particle notion" exists in limited circumstances.
Haag: The "quantum …eld" concept is the encorporation
of the principle of locality into quantum theory.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 15 / 16
34. The Quantum Field and Particles
The Relativity Principle: All observers are equivalent,
including non-inertial and inertial.
Consequences:
No "universal" de…nition of particles and No
"universal" de…nition of vacuum
Need observer dependent de…nition of particles and
vacuums state.
The …eld is more fundamental than particles:
The "particle notion" exists in limited circumstances.
Haag: The "quantum …eld" concept is the encorporation
of the principle of locality into quantum theory.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 15 / 16
35. The Quantum Field and Particles
The Relativity Principle: All observers are equivalent,
including non-inertial and inertial.
Consequences:
No "universal" de…nition of particles and No
"universal" de…nition of vacuum
Need observer dependent de…nition of particles and
vacuums state.
The …eld is more fundamental than particles:
The "particle notion" exists in limited circumstances.
Haag: The "quantum …eld" concept is the encorporation
of the principle of locality into quantum theory.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 15 / 16
36. The Quantum Field and Particles
The Relativity Principle: All observers are equivalent,
including non-inertial and inertial.
Consequences:
No "universal" de…nition of particles and No
"universal" de…nition of vacuum
Need observer dependent de…nition of particles and
vacuums state.
The …eld is more fundamental than particles:
The "particle notion" exists in limited circumstances.
Haag: The "quantum …eld" concept is the encorporation
of the principle of locality into quantum theory.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 15 / 16
37. The Quantum Field and Particles
The Relativity Principle: All observers are equivalent,
including non-inertial and inertial.
Consequences:
No "universal" de…nition of particles and No
"universal" de…nition of vacuum
Need observer dependent de…nition of particles and
vacuums state.
The …eld is more fundamental than particles:
The "particle notion" exists in limited circumstances.
Haag: The "quantum …eld" concept is the encorporation
of the principle of locality into quantum theory.
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 15 / 16
38. Comments/Open Questions
1 This type of construction only works for free …elds.
1 Free …elds are linear …elds, for nonlinear …elds no Ω
exists on S.
2 Examples of nonlinear …elds: interacting …elds, gauge
…elds, and gravity
3 Renormalization in curved space-time has been
formulated by Fredenhagen-Brunetti and Hollands-Wald.
2 It does not consider back reaction:
Gµν = 8π ˆTµν
I.e. the e¤ect on geometry by these …elds.
IWe treat the geometry as …xed when it isn’t really.
3 The BIG question: quantum gravity?
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16
39. Comments/Open Questions
1 This type of construction only works for free …elds.
1 Free …elds are linear …elds, for nonlinear …elds no Ω
exists on S.
2 Examples of nonlinear …elds: interacting …elds, gauge
…elds, and gravity
3 Renormalization in curved space-time has been
formulated by Fredenhagen-Brunetti and Hollands-Wald.
2 It does not consider back reaction:
Gµν = 8π ˆTµν
I.e. the e¤ect on geometry by these …elds.
IWe treat the geometry as …xed when it isn’t really.
3 The BIG question: quantum gravity?
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16
40. Comments/Open Questions
1 This type of construction only works for free …elds.
1 Free …elds are linear …elds, for nonlinear …elds no Ω
exists on S.
2 Examples of nonlinear …elds: interacting …elds, gauge
…elds, and gravity
3 Renormalization in curved space-time has been
formulated by Fredenhagen-Brunetti and Hollands-Wald.
2 It does not consider back reaction:
Gµν = 8π ˆTµν
I.e. the e¤ect on geometry by these …elds.
IWe treat the geometry as …xed when it isn’t really.
3 The BIG question: quantum gravity?
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16
41. Comments/Open Questions
1 This type of construction only works for free …elds.
1 Free …elds are linear …elds, for nonlinear …elds no Ω
exists on S.
2 Examples of nonlinear …elds: interacting …elds, gauge
…elds, and gravity
3 Renormalization in curved space-time has been
formulated by Fredenhagen-Brunetti and Hollands-Wald.
2 It does not consider back reaction:
Gµν = 8π ˆTµν
I.e. the e¤ect on geometry by these …elds.
IWe treat the geometry as …xed when it isn’t really.
3 The BIG question: quantum gravity?
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16
42. Comments/Open Questions
1 This type of construction only works for free …elds.
1 Free …elds are linear …elds, for nonlinear …elds no Ω
exists on S.
2 Examples of nonlinear …elds: interacting …elds, gauge
…elds, and gravity
3 Renormalization in curved space-time has been
formulated by Fredenhagen-Brunetti and Hollands-Wald.
2 It does not consider back reaction:
Gµν = 8π ˆTµν
I.e. the e¤ect on geometry by these …elds.
IWe treat the geometry as …xed when it isn’t really.
3 The BIG question: quantum gravity?
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16
43. Comments/Open Questions
1 This type of construction only works for free …elds.
1 Free …elds are linear …elds, for nonlinear …elds no Ω
exists on S.
2 Examples of nonlinear …elds: interacting …elds, gauge
…elds, and gravity
3 Renormalization in curved space-time has been
formulated by Fredenhagen-Brunetti and Hollands-Wald.
2 It does not consider back reaction:
Gµν = 8π ˆTµν
I.e. the e¤ect on geometry by these …elds.
IWe treat the geometry as …xed when it isn’t really.
3 The BIG question: quantum gravity?
P. Tillman (Queensborough Community College)The Origin of Complex Structures in QFT on Curved Space-Times01/08/08 16 / 16