July 30, 2013

Dot Products in Special Relativity

Sabrina Gonzalez Pasterski

I present a derivation of the Minkowski metric and visualizations of the space-time dot product.

Dot Products in Special Relativity

Sabrina Gonzalez Pasterski

I present a derivation of the Minkowski metric and visualizations of the space-time dot product.

August 1, 2013

Degeneracy vs. Energy Level Scaling for Hydrogen

Sabrina Gonzalez Pasterski

I explore the relationship between degeneracy and energy for the Hydrogen atom.

Degeneracy vs. Energy Level Scaling for Hydrogen

Sabrina Gonzalez Pasterski

I explore the relationship between degeneracy and energy for the Hydrogen atom.

September 8, 2013

Wavefunctions and the Hamilton-Jacobi Equation

Sabrina Gonzalez Pasterski

I show how the differential equation governing a distribution over classical trajectories is consistent with the quantum Schrodinger Equation in the h --> 0 limit.

Wavefunctions and the Hamilton-Jacobi Equation

Sabrina Gonzalez Pasterski

I show how the differential equation governing a distribution over classical trajectories is consistent with the quantum Schrodinger Equation in the h --> 0 limit.

March 2, 2014

Covariant Derivatives and the Hamilton-Jacobi Equation

Sabrina Gonzalez Pasterski

I define a covariant derivative to simplify how higher order derivatives act on a classical generating function.

Covariant Derivatives and the Hamilton-Jacobi Equation

Sabrina Gonzalez Pasterski

I define a covariant derivative to simplify how higher order derivatives act on a classical generating function.

March 2, 2014

Expectation Values from the Hamilton-Jacobi Equation

Sabrina Gonzalez Pasterski

I use the classical Hamilton-Jacobi Equation to formulate a definition for operator expectation values consistent with those for quantum operators.

Expectation Values from the Hamilton-Jacobi Equation

Sabrina Gonzalez Pasterski

I use the classical Hamilton-Jacobi Equation to formulate a definition for operator expectation values consistent with those for quantum operators.

March 17, 2014

Fields with a Gaussian Measure

Sabrina Gonzalez Pasterski

A calculation.

Fields with a Gaussian Measure

Sabrina Gonzalez Pasterski

A calculation.

August 4, 2013

Conformal Mapping of Displacement Vectors

Sabrina Gonzalez Pasterski

I explore the asymmetry between displacements in space and time for motion in one dimension.

Conformal Mapping of Displacement Vectors

Sabrina Gonzalez Pasterski

I explore the asymmetry between displacements in space and time for motion in one dimension.

August 2, 2013

An Alternate Approach for Finding an Orthogonal Basis

Sabrina Gonzalez Pasterski

I present an alternative to the Gram-Schmidt method for finding a basis of orthogonal vectors spanning the same space as a set of starting vectors.

An Alternate Approach for Finding an Orthogonal Basis

Sabrina Gonzalez Pasterski

I present an alternative to the Gram-Schmidt method for finding a basis of orthogonal vectors spanning the same space as a set of starting vectors.

August 5, 2013

Space and Time

Sabrina Gonzalez Pasterski

I arrive at Maxwell’s Equations, gauge invariance, and the force law for charges, starting from a conserved current.

Space and Time

Sabrina Gonzalez Pasterski

I arrive at Maxwell’s Equations, gauge invariance, and the force law for charges, starting from a conserved current.

September 8, 2013

Solving the Shrodinger Equation Using a Complex Gauge

Sabrina Gonzalez Pasterski

I show a quick route to the n = l + 1 wave functions and energies for the Hydrogen atom using a method which is also applied to find the ground state of the simple harmonic oscillator.

Solving the Shrodinger Equation Using a Complex Gauge

Sabrina Gonzalez Pasterski

I show a quick route to the n = l + 1 wave functions and energies for the Hydrogen atom using a method which is also applied to find the ground state of the simple harmonic oscillator.

June 12, 2014 (JHEP on August 11, 2014)

Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix

D. Kapec, V. Lysov, S. Pasterski and A. Strominger

It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a Virasoro symmetry which acts on the conformal sphere at null infinity.

Semiclassical Virasoro Symmetry of the Quantum Gravity S-Matrix

D. Kapec, V. Lysov, S. Pasterski and A. Strominger

It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a Virasoro symmetry which acts on the conformal sphere at null infinity.

April 20, 2014

Gaussian Measures and the QM Oscillator

Sabrina Gonzalez Pasterski

In this paper, I show how probability densities associated with a Gaussian field can be expressed in terms of the Boltzmann heat kernel.

Gaussian Measures and the QM Oscillator

Sabrina Gonzalez Pasterski

In this paper, I show how probability densities associated with a Gaussian field can be expressed in terms of the Boltzmann heat kernel.

March 31, 2014

Soft Theorems and Symmetry

Sabrina Gonzalez Pasterski

I examine the connection between symmetries and soft factors for the case of graviton scattering.

Soft Theorems and Symmetry

Sabrina Gonzalez Pasterski

I examine the connection between symmetries and soft factors for the case of graviton scattering.

March 10, 2014

Gaussian Measures and Commutators

Sabrina Gonzalez Pasterski

I show how defining a field with a gaussian measure gives rise to a classical interpretation of the equal time commutation relations for a field and its derivative.

Gaussian Measures and Commutators

Sabrina Gonzalez Pasterski

I show how defining a field with a gaussian measure gives rise to a classical interpretation of the equal time commutation relations for a field and its derivative.

March 31, 2014

From Gaussian Measures to SDE

Sabrina Gonzalez Pasterski

In this paper, I show how a field with gaussian measure can give the Schwinger-Dyson Equations of QFT in Wick rotated time.

From Gaussian Measures to SDE

Sabrina Gonzalez Pasterski

In this paper, I show how a field with gaussian measure can give the Schwinger-Dyson Equations of QFT in Wick rotated time.

July 14, 2013

Hamilton’s Equations of Motion

Sabrina Gonzalez Pasterski

I motivate Hamilton’s equations of motion using a geometrical picture of contours in phase space. The following considers a single cartesian coordinate x with conjugate momentum p.

Hamilton’s Equations of Motion

Sabrina Gonzalez Pasterski

I motivate Hamilton’s equations of motion using a geometrical picture of contours in phase space. The following considers a single cartesian coordinate x with conjugate momentum p.

July 23, 2013

Visualization for Spin-1/2 Inner Products

Sabrina Gonzalez Pasterski

I present a geometrical visualization for the magnitude of the inner product of two spin-1/2 states.

Visualization for Spin-1/2 Inner Products

Sabrina Gonzalez Pasterski

I present a geometrical visualization for the magnitude of the inner product of two spin-1/2 states.

July 24, 2013

Motivating Special Relativity using Linear Algebra

Sabrina Gonzalez Pasterski

I present a geometrical derivation of results from Special Relativity for a single spatial coordinate.

Motivating Special Relativity using Linear Algebra

Sabrina Gonzalez Pasterski

I present a geometrical derivation of results from Special Relativity for a single spatial coordinate.

July 27, 2013

Position as a Single-Valued Function of Time

Sabrina Gonzalez Pasterski

I present a geometrical illustration of how the speed limit in Special Relativity requires position to be a single-valued function of time in any reference frame.

Position as a Single-Valued Function of Time

Sabrina Gonzalez Pasterski

I present a geometrical illustration of how the speed limit in Special Relativity requires position to be a single-valued function of time in any reference frame.

July 30, 2013

The Schrodinger Equation and Phase Space

Sabrina Gonzalez Pasterski

I arrive at the Schrodinger Equation given a particular form for probabilities in phase space.

The Schrodinger Equation and Phase Space

Sabrina Gonzalez Pasterski

I arrive at the Schrodinger Equation given a particular form for probabilities in phase space.

February 28, 2014

Insights into Becoming a Theoretical Physicist

Sabrina Gonzalez Pasterski

An inspirational speech given in Princeton, NJ.

The $500 honorarium was later used to pay the author's fee on the September 12th Physical Review Letter Editors' Suggestion.

Insights into Becoming a Theoretical Physicist

Sabrina Gonzalez Pasterski

An inspirational speech given in Princeton, NJ.

The $500 honorarium was later used to pay the author's fee on the September 12th Physical Review Letter Editors' Suggestion.

April 21, 2014

Gaussian Measures in Quantum Theory

Sabrina Gonzalez Pasterski

Invited talk. Harvard String Group.

Gaussian Measures in Quantum Theory

Sabrina Gonzalez Pasterski

Invited talk. Harvard String Group.

July 14, 2014 (PRL Editors' Suggestion on September 12, 2014)

Low’s Subleading Soft Theorem as a Symmetry of QED

V. Lysov, S. Pasterski and A. Strominger

It was shown by F. Low in the 1950s that the subleading terms of soft photon S-matrix elements obey a universal linear relation. In this paper we give a new interpretation to this old relation, for the case of massless QED, as an infinitesimal symmetry of the S-matrix.

Low’s Subleading Soft Theorem as a Symmetry of QED

V. Lysov, S. Pasterski and A. Strominger

It was shown by F. Low in the 1950s that the subleading terms of soft photon S-matrix elements obey a universal linear relation. In this paper we give a new interpretation to this old relation, for the case of massless QED, as an infinitesimal symmetry of the S-matrix.

July 31, 2014

Classical Interpretation of the Weinberg Soft Factor

Sabrina Gonzalez Pasterski

I show how the radiation emitted during the scattering of non-relativistic charged particles corresponds to the O(1/w) soft factor in QED.

Classical Interpretation of the Weinberg Soft Factor

Sabrina Gonzalez Pasterski

I show how the radiation emitted during the scattering of non-relativistic charged particles corresponds to the O(1/w) soft factor in QED.

September 10, 2014

Generalizing the Soft Factor/Classical Connection

Sabrina Gonzalez Pasterski

I show that the polarization vector times the soft factor for a photon emitted in the xˆ direction is proportional to the time integral of the classical radiated electric field for the same scattering process, as measured by an observer sitting along that direction far away.

Generalizing the Soft Factor/Classical Connection

Sabrina Gonzalez Pasterski

I show that the polarization vector times the soft factor for a photon emitted in the xˆ direction is proportional to the time integral of the classical radiated electric field for the same scattering process, as measured by an observer sitting along that direction far away.

September 11, 2014

Subtleties of Zero Modes

Sabrina Gonzalez Pasterski

I introduce the two distinct zero mode components of the gauge field expansion and the subtlety of distinguishing their commutation relations.

Subtleties of Zero Modes

Sabrina Gonzalez Pasterski

I introduce the two distinct zero mode components of the gauge field expansion and the subtlety of distinguishing their commutation relations.

December 16, 2014

Speed Demon

Sabrina Gonzalez Pasterski

I propose a detector arrangement/measurement corresponding to the subleading soft graviton theorem.

Speed Demon

Sabrina Gonzalez Pasterski

I propose a detector arrangement/measurement corresponding to the subleading soft graviton theorem.

September 13, 2014

Massive Soft Factors

Sabrina Gonzalez Pasterski

I show that the soft factor corresponds to the measured time integrated radiation for any speed.

Massive Soft Factors

Sabrina Gonzalez Pasterski

I show that the soft factor corresponds to the measured time integrated radiation for any speed.

November 20, 2014

Gravitational Memory, BMS Supertranslations and Soft Theorems

A. Strominger and A. Zhiboedov

My 'Massive Soft Factors' and 'Generalizing the Soft Factor/Classical Connection' papers were reviewed by, discussed by, and expanded upon by my fellow group members in this most interesting paper! I am honored to be mentioned.

Gravitational Memory, BMS Supertranslations and Soft Theorems

A. Strominger and A. Zhiboedov

My 'Massive Soft Factors' and 'Generalizing the Soft Factor/Classical Connection' papers were reviewed by, discussed by, and expanded upon by my fellow group members in this most interesting paper! I am honored to be mentioned.

Physical Review Letters Editors' Suggestion Volume 113, Issue 11

Soft Physics

ISBN: 9780986368592

February x, 2015

New Gravitational Memories

S. Pasterski, A. Strominger and A. Zhiboedov

New Gravitational Memories

S. Pasterski, A. Strominger and A. Zhiboedov

February xx, 2015

New Gravitational Memories

Sabrina Gonzalez Pasterski

Invited talk. Harvard String Family.

New Gravitational Memories

Sabrina Gonzalez Pasterski

Invited talk. Harvard String Family.

March 20, 2015

tba

Sabrina Gonzalez Pasterski

Invited talk. Fajardo, Puerto Rico.

tba

Sabrina Gonzalez Pasterski

Invited talk. Fajardo, Puerto Rico.

2015 single-author notes to be cited as ISBN: 9780986368585

2013-2014 single-author notes are cited as ISBN: 9780986368592