Richard Feynman's Functional Integral 1 - Action
By Chris Austin | July 27th 2012 12:38 AM | 7 comments | Print | E-mail | Track Comments

I work independently on high-energy physics theory, in Maryport on the north-west coast of England. I have also developed View Chris's Profile

Hello, and welcome to this new blog. I am Chris Austin, I work independently on high energy physics theory, in Maryport on the north-west coast of England.

With regard to the title of the blog, may I first apologize to Lewis Carroll for the take-off.

I am working on models of high energy physics in which the known elementary particles, which comprise 6 types of quark triplet, plus the electron and 2 similar heavier particles, plus 3 types of neutrino, plus the photon and 11 roughly similar particles that transmit forces, plus a Higgs boson candidate, plus the graviton, arise from supergravity in 11 dimensions, otherwise known as -theory, when the 7 extra dimensions of space are curled up into a suitable small shape. The 7 extra dimensions of space are too small to have been seen up to now, but could become visible for the first time at the Large Hadron Collider, which started operation at CERN, in the region of the French/Swiss border near Geneva, in 2009.

While working on these problems, I often feel as though I am crawling on hands and knees through an unfamiliar but wonderful place.

-theory and superstring theory have taken a few knocks in recent years, and over the course of some posts here, I'll try to say a few words for the defence, while possibly also filling in a bit of the background.

I hope this blog will be fun and useful for everyone with an interest in science, so although I'll pop up a few formulae, I'll try to make them friendly by explaining all the pieces. Please feel free to ask a question in the Comments, if you think anything in the post is unclear.

Today I would like to tell you about the foundation of our understanding of high energy physics, which is Richard Feynman's functional integral, but first I have to tell you about action.

The post got a bit long, so I have divided it into ten parts. The following parts, which will appear at intervals of about a month, are Multiple Molecules, Electromagnetism, Action for Fields, Radiation in an Oven, Matrix Multiplication, The Functional Integral, Gauge Invariance, Photons, and Interactions.

About 60 years after Sir Isaac Newton published his laws of motion in 1687, Pierre-Louis Moreau de Maupertuis discovered that Newton's laws follow from the requirement that a quantity he called "action", that depends on the positions and motions of a collection of objects over a period of time, should be relatively unaltered by small changes in the positions and motions of those objects.

For an example, let's consider a collection of objects, such that each object behaves approximately as though its mass is concentrated at a single point, the objects are moving slowly compared to the speed of light, and the forces between the objects arise from a contribution to their total energy, called their potential energy , that depends on their positions but not on their motions. This is a useful first approximation for many things in the everyday world, for the Sun and the planets in the Solar System, for the motions of most of the stars in our galaxy, and for the atoms in a solid, liquid, gas, or living thing.

If denotes the sum of the kinetic energies of the objects, or in other words, the energy due to their motion, where the kinetic energy of a pointlike object of mass moving at a speed is if is small compared to the speed of light, then the formula for the action of the objects over a period of time that begins at and ends at is:

Here is the action. The symbol was once an alternative form of the letter , and is nowadays called the integral sign. It means a sum of what follows it. The upright letter in means a tiny amount of,'' and was introduced in this context by Gottfried Wilhelm Leibniz. The idea is that the period from to is divided up into a great number of tiny time intervals, each so small that and only change by a tiny amount during any one time interval. The notation means that is the sum of a contribution from each of these tiny time intervals, and the contribution from a tiny time interval of length is the product of and the tiny amount of time , where and are evaluated at an arbitrary moment during that interval. Then for motions such that and vary smoothly with time, the sum of all the contributions changes by an ever smaller amount as the period from to is divided up ever more finely, and is the limit of the sum of all the contributions, as the period is divided up so finely that the length of the longest tiny interval tends to 0.

We can specify the position of an object at a particular moment in time by 3 numbers, called coordinates. For example, we can specify the position of an aeroplane by its latitude and longitude, in degrees, and its altitude, in meters. We can write these 3 numbers as a list, for example (latitude, longitude, altitude).

If the number of objects in our example is , then their positions, at a particular moment in time, can be specified by a table of numbers with 3 columns and rows. Each row gives the coordinates of a different one of the objects. The motions of the objects over a period of time can be represented as accurately as desired by a series of such tables, one for each of a closely spaced sequence of moments in time.

It is convenient to let a single symbol, say , represent this entire collection of data. Then if the letter , for example, represents one of the numbers 1, 2, 3, and the letter , for example, represents one of the numbers , the value of position coordinate number , of object number , at time , can be represented as , where the subscripts , , and are called indexes. Alternative notations for the same thing, which can be used as convenient, are , and , for example.

The speed of an object is proportional to the rate at which its coordinates change with time. If a collection of data that gives the value of a quantity at each moment in time is represented by a symbol or expression , then the collection of data that gives the rate of change of that quantity with time, at each moment in time, is often represented as , or alternatively . The Leibniz in the numerator of means, the change in the following expression, when the time changes by the tiny amount ,'' and indicates that the formula is to be taken in the limit where the size of the time interval tends to 0. The value of at time is , which is well-defined if changes smoothly with . The coordinates of the objects in our example will change smoothly with for a sensible choice of position coordinates, since we assumed that is small compared to the speed of light, and thus finite.

To calculate the speed of object number , we need to know to the distance travelled for a small change in its coordinates . For example the distance travelled by an aeroplane, for a change in its longitude at fixed latitude and altitude, is smaller, the closer the aeroplane is to the north or south pole.

For the flat 2-dimensional world of Euclidean geometry, we can choose as coordinates the distances from 2 fixed straight lines, at right-angles to one another. These coordinates were introduced by René Descartes, and are called Cartesian coordinates. The distance between two points whose coordinates differ by and whose coordinates differ by is then given by Pythagoras as , since directly from the diagram, .

For the flat 3-dimensional generalization of Euclidean geometry, called 3-dimensional Euclidean space, we can choose as coordinates the distances from 3 fixed flat planes, each at right-angles to the other two. These are also called Cartesian coordinates. The distance between two points whose coordinates differ by , for and , is then , which follows from applying Pythagoras first to the and coordinates, and then to and .

The assumptions of our example imply that any gravitational fields present are sufficiently weak that if we use Cartesian coordinates, then distances are given by Pythagoras to a good approximation, for otherwise the objects would not continue to move slowly compared to the speed of light. Then in Cartesian coordinates, the speed of object number is , and the sum of the kinetic energies of the objects is given by:

The symbol is the upper-case Greek letter Sigma, and indicates a sum of what follows it. The idea is that each contribution to the sum is obtained from the expression that follows the , by substituting a specific value for one of the indexes in the expression, and the notations below and above the show which index is to be substituted, and the range of values of that index, for which terms are to be included in the sum. Thus the meaning of is quite similar to the meaning of as above. The difference is that is used for a sum over a discrete index such as or , while , together with a tiny factor such as , is used for a sum over a continuous index such as .

Let's now consider a small change to the positions and motions of the objects during the period of time from to . I'll represent the change, or perturbation'', of the positions and motions by the Greek letter , pronounced epsilon, which is often used to represent a small quantity, so the modified positions and motions are represented by . Here , like , represents an entire collection of data, for example it could introduce different types of wobbles to the motions of each of the objects. I shall assume that is 0 at and , or in other words, that for all values of and all values of , while for times between and , I shall assume only that all the are small, and change smoothly with time.

Near the start of the post, above, I said that de Maupertuis's requirement, which implies Newton's laws, is that the action should be relatively unaltered by small changes to the positions and motions of the objects. What I meant by that is that as tends to 0, or in other words, as approaches 0 for all relevant values of , , and , the change to the action should tend to 0 more rapidly than in proportion to .

The change to the contribution to the action, that results from the replacement of by , is:

since , and the contribution from the first expression, usually called a term, in the right-hand side of this, cancels against the last expression in the left-hand side of the above equation for each value of the indexes and , while the third term, proportional to , is much smaller than the second term for very small , so can be neglected. The symbol means approximately equal to''.

Let's now consider the rate of change with time of a product , where and represent collections of data that give the values, at each moment in time, of quantities that change smoothly with time. The expression represents the collection of data that gives the value of the product at each moment in time, so from above, the collection of data that gives the rate of change with time of the product , at each moment in time, can be represented as . And:

The second line here follows from noting that if is any expression that varies smoothly with , then since , we have . The third line follows because the last contribution in the second line cancels part of the contribution before it, and the ratio is 0 in the limit where tends to 0.

The above formula is true at every value of the time , so it can be summarized as:

This is called Leibniz's rule for the rate of change of a product.

Applying this to the product , we have:

From this result and the previous one, the change to that results from the replacement of by , is:

From now on, if an expression represents a collection of data that gives the value of a quantity at each moment in time, I shall for brevity just say that is a time-dependent quantity.

Let's now consider the expression , where is any time-dependent quantity whose value changes smoothly with time. From the description I gave near the start of the post above, this expression is given by dividing the period from to up into a great number of tiny time intervals, each so small that only changes by a tiny amount during any one time interval, and adding together a contribution from each of these tiny time intervals. The contribution from a tiny time interval of length is , where is evaluated at an arbitrary moment during that interval, and the expression is the limit of the sum of all the contributions, as the period is divided up so finely that the length of the longest tiny interval tends to 0.

For a tiny time interval that starts at time and finishes at time , where , we have in the limit where tends to 0, so the contribution of that interval is . When we add together the contributions of all the tiny intervals, the term in the contribution of each interval except the last one cancels the term in the contribution of next interval, so that:

In words, this means that the integral of the rate of change of a quantity is equal to the net change of that quantity.

This is true, in particular, if is , so from the previous result, the change to that results from the replacement of by , is:

However we assumed above that , which is a small modification to the positions and motions of the objects, is 0 at and . So the first two terms inside the outer two pairs of parentheses in the above expression are 0, so the change to that results from the replacement of by , is:

The remaining contribution to the change of the action in our example, that results from modifying the positions and motions of the objects by the small perturbation , is the contribution from the change of , where is the potential energy, which depends on the positions of the objects, but not on their motions. The symbol represents the collection of data that includes the value of the potential energy at each moment in time, and the value of the potential energy at time , which we can write as or as convenient, depends on the values of the coordinates at time , but not on the values of the coordinates at any other time. For example if the objects are stars or planets then the significant potential energy is their gravitational potential energy, given by:

where is the distance between object and object , and is Newton's constant of gravitation.

If a quantity, such as the potential energy, depends on a number of quantities that can vary continuously, where represents the collection of those quantities, and the index distinguishes the quantities in the collection, and if the collection of data that gives the value of the dependent quantity at each , or in other words, at each set of values of the quantities , is represented by a symbol , then the collection of data that gives the rate of change of the dependent quantity as the quantity changes, while all the other quantities in have fixed values, is usually represented as , or alternatively as . The symbol is an alternative notation for Leibniz's , and , where the quantities in the collection other than all have the same values in both terms in the numerator in the right-hand side as they have in the left-hand side, so their values don't need to be displayed.

If the value changes smoothly with , or in other words, smoothly with the quantities in the collection , then for near a reference collection , in the sense that all the quantities are small in magnitude, the value can be represented approximately as:

where as the magnitudes of all the tend to 0, the error of this approximate representation tends to 0 more rapidly than in proportion to those magnitudes. For the two sides of the above formula are equal when . And by using the above definition of for the case when is , we find that , where is the Greek letter delta, and , which is called the Kronecker delta after Leopold Kronecker, is 1 when , and 0 otherwise. Thus applying to the right-hand side of the above formula gives for all , which is in agreement with the application of to the left-hand side when . Thus the two sides of the above formula would be in agreement for all if the quantities were independent of . This is not so in general, but the assumption that changes smoothly with implies that the differences tend to 0 at least as fast as the differences as approaches , so the error of the above formula tends to 0 at least as fast as products of two of those differences, and thus more rapidly than in proportion to those differences.

Applying the above formula to the potential energy, with taken as , and taken as , we have:

where the error of this formula tends to 0 more rapidly than in proportion to , as tends to 0 for all relevant values of and .

Combining this formula for the change of the potential energy with the formula for the change of the kinetic energy we obtained before it, we find that the change to the action that results from the replacement of by , is:

where the error of this formula tends to 0 more rapidly than in proportion to , as tends to 0.

De Maupertuis's principle requires that the change to the action should tend to 0 more rapidly than in proportion to , as tends to 0. But from the above formula, this is only possible for all perturbations such that is 0 at and , and all the change smoothly with time, if:

for all relevant values of , , and .

We are using Cartesian coordinates, so is the 'th component of the velocity of the 'th object, and , which is usually written as , is the 'th component of the acceleration of the 'th object. And by the definition of potential energy, the 'th component of the force on the 'th object is . Thus the above equation is Newton's second law of motion.

Let's now consider the rate of change with time of the total energy , when the objects move in accordance with Newton's second law of motion, which we have just derived from de Maupertuis's principle. From Leibniz's rule for the rate of change of a product, which we proved above, the rate of change of is , so the rate of change with time of the kinetic energy is:

And by choosing to be , in the formula we derived above, for the change of the potential energy when is replaced by , we obtain the rate of change with time of the potential energy as:

since only depends on time through the dependence on time of the coordinates of the objects.

From the sum of the above two formulae, we obtain the rate of change with time of the total energy of the objects, when they move in accordance with Newton's second law of motion, as:

Thus the total energy of the objects never changes. This is usually referred to as the conservation of total energy. (The everyday phrase, conservation of energy,'' refers to trying to reduce the rate at which some particular forms of energy, such as the potential energy associated with the arrangement of the atoms within the lattice structure or molecules of a chemical fuel, are converted to other forms of energy.)

### Stationary action for a planet

To illustrate the practical application of de Maupertuis's discovery, which is sometimes called the principle of stationary action, let's consider a planet in orbit around the Sun, neglecting the gravitational effects of the other planets, which are relatively small. The mass of the Sun is much greater than the mass of the planet, so to a good approximation, we can treat the Sun as fixed in position, and just consider the motion of the planet around the Sun. The gravitational force on the planet is always in the direction of the straight line from the planet to the Sun, so the planet stays in the 2-dimensional plane defined by the straight line from the planet's initial position to the Sun, and the direction of the planet's initial velocity, which I shall assume is not exactly along that line.

It is convenient to specify the planet's position in this plane by the distance from the planet to the Sun, and the angle between the straight line from the planet to the Sun, and the initial direction of that line. I shall represent that angle by , which is the Greek letter theta. To keep the formulae as simple as possible, the angle will be measured not in degrees but in "radians", where 1 radian is the angle turned through when something moving along a circular path has travelled a distance along the circle equal to the radius of the circle. Thus a full rotation is radians, and 1 radian is approximately .

Due to measuring the angle in radians, the distance travelled by the planet when increases by a small amount at fixed is , and in the limit when tends to 0, this is in the direction perpendicular to the straight line from the planet to the Sun. Thus the square of the planet's speed can be calculated from and using Pythagoras, so the kinetic energy of the planet is . The gravitational potential energy is , so the action is:

From above, the total energy:

is independent of time, and thus partly characterizes the planet's orbit. De Maupertuis's principle leads to one independent equation of motion for each coordinate of each moving object. We have already obtained the time-independence of the total energy from one combination of the equations of motion, so we only need to obtain one of the two equations of motion directly by requiring that the action is relatively unaltered by a small modification to the time dependence of the coordinates. Using the same method as above, we find that if a small time-dependent perturbation is added to , such that depends smoothly on , and , then the modification to the action is:

where the error of this formula is proportional to , and thus tends to 0 more rapidly than in proportion to , as tends to 0. To obtain this formula, we used the equality of the integral of the rate of change and the net change, as above, applied to the expression , together with Leibniz's rule for the rate of change of a product, as above, applied to the product of and .

De Maupertuis's principle requires that the change to the action should tend to 0 more rapidly than in proportion to , as tends to 0. But from the above formula, this is only possible for all perturbations such that is 0 at and , and changes smoothly with time, if , for all relevant values of . This means that is independent of time.

For a tiny amount of time , the area swept out by the straight line from the Sun to the planet during the time interval is approximately , which is the area of the right-angled triangle made by the straight lines from the Sun to the planet at the times and , together with the straight line tangential to the circle of radius centred at the Sun, that meets that circle at the position of the planet at time . The difference between , and the area swept out by the straight line from the Sun to the planet during the time interval , tends to 0 in proportion to as tends to 0, and thus more rapidly than in proportion to , so the rate at which the straight line from the Sun to the planet sweeps out area is . We found above from de Maupertuis's principle that this is independent of time, so the straight line from the Sun to the planet sweeps out equal areas in equal times. This is the second of the three laws of planetary motion, which Johannes Kepler discovered by studying the astronomical measurements made by Tycho Brahe.

The product , which is also independent of time since the planet's mass is constant, is called the orbital angular momentum of the planet. I shall represent it by . The value of , like the value of the planet's total energy, partly characterizes the orbit of the planet.

To find the possible shapes of the planet's orbit, we can convert the rate of change of with time to the rate of change of with , using the relation . The time interval during which changes by a tiny amount is , so . Using this result and also the relation in the above formula for the planet's total energy , we find:

Rearranging this formula, we find:

To use this formula to find the possible orbits of the planet, it is helpful to know about the Cartesian coordinates of something moving around a circle, and their rate of change with angle. If something is moving along a circular path, and is the angle in radians, as above, between the straight line from the centre of the moving object to the centre of the circle, and a fixed straight line in the plane of the circle though the centre of the circle, then the traditional names for the Cartesian coordinates of the centre of the moving object, relative to the centre of the circle, in units of the radius of the circle, are for the coordinate parallel to the fixed straight line, and for the coordinate perpendicular to the fixed straight line in the plane of the circle. The directions of the coordinates are chosen so that and . From Pythagoras, we have , for all .

If the object starts at angle and goes round the circle times, so that increases by , where is any whole number, then the Cartesian coordinates of the centre of the object come back to their initial values, so that , and , for all whole numbers . This diagram shows , for in the range to .

When the angle increases by a tiny amount , the changes to the coordinates of the centre of the object are approximately the same as they would be if the object moved a distance along the straight line tangential to the circle at instead of exactly along the circle, where is the radius of the circle, and the relative error of this approximation tends to 0 as tends to 0. And from this diagram, the change to the Cartesian coordinate parallel to the fixed straight line, when the centre of the object moves a distance along the tangential straight line in the direction of increasing , is , and the change to the Cartesian coordinate perpendicular to the fixed straight line is . Thus:

and

Returning to the planet in orbit around the Sun, we can now confirm that the above formula for implies that the orbit of the planet is an ellipse with the Sun at one focus, in agreement with Kepler's first law of planetary motion. An ellipse is the curve formed by all the points in a plane such that the sum of the distances from a point on the ellipse to two fixed points, called the focuses of the ellipse, has a fixed value. With the Sun at one focus, the distance from the planet to that focus is . I shall represent the distance between the two focuses by , and choose the fixed direction corresponding to to be along the straight line from the Sun to the other focus. Then using Cartesian coordinates for a moment, the Cartesian coordinates of the other focus are and the Cartesian coordinates of the planet are , so by Pythagoras, the distance from the planet to the second focus is:

I shall represent the fixed sum of the distances from the planet to the two focuses by , so if the planet's orbit is an ellipse characterized by the distances and , then and are related by:

Rearranging this formula, we find:

We can calculate the rate of change of with from this formula by using Leibniz's rule for the rate of change of a product, which we obtained above, since the product is constant, so its rate of change is 0. Thus:

So we find:

To compare this formula for for an ellipse with the formula for for the planet's orbit that we obtained above from de Maupertuis's principle, we square both sides of the ellipse formula, and then use the above relation between and for the ellipse to express the right-hand side in terms of instead of :

This exactly matches the formula for for the planet's orbit that we obtained above from de Maupertuis's principle, if and , so that , and , where denotes the absolute value of . The value of is negative because the planet is gravitationally bound to the Sun.

The time taken for the planet to complete one orbit is called the orbital period, and I shall represent it by . We found above that the rate at which the straight line from the Sun to the planet sweeps out area is . Thus

where represents the area enclosed by the orbit. To calculate , it is helpful to use Cartesian coordinates centred at the centre of the ellipse, halfway between the two focuses. Then with representing the coordinate parallel to the line between the two focuses and representing the coordinate perpendicular to this line in the plane of the ellipse, the distances from the point to the focuses of the ellipse are and by Pythagoras, so the equation of the ellipse is:

Squaring both sides and rearranging, this becomes:

Squaring both sides of this and rearranging, it becomes:

If we rewrite this in terms of a rescaled coordinate such that , it becomes the equation of a circle of radius , with area . Thus the actual area of the ellipse is:

Thus . We found above that , so:

is the length of the major axis of the ellipse, so this shows that the square of the orbital period is equal to the cube of the length of the major axis of the orbit, multiplied by a quantity that is the same for all the planets. This is Kepler's third law of planetary motion.

The clue that led to the discovery of quantum mechanics, whose principles are summarized in Feynman's functional integral, came from the attempted application to electromagnetic radiation of discoveries about heat and temperature. In the next part of this post, Multiple Molecules, we'll look at some of those discoveries.

It¨s a good work, but it is not all mathematics, the imagination is very important.

No, it isn't.  Too many people are putting the imaginary first and never learn the science (or math). Einstein did not get published based on his imagination.
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I appreciate the point Carlos was making. In my work, I have often found that imagining a solution to a problem, initially in a somewhat vague way, is a good first step to developing a concrete solution. After that comes developing a more detailed description of the imagined or conjectured solution, and after that, expression of the description in a mathematical form.

Here are two comments by Einstein on imagination:

“I am enough of an artist to draw freely upon my imagination. Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.” ― Albert Einstein

“Logic will get you from A to Z; imagination will get you everywhere.” ― Albert Einstein
Dear Chris, What a wonderful idea! Thank you for explaining the important concept of the action in physics so well. We need more such explanations of physical-mathematical ideas that often are only vaguely understood by many informed readers. I look forward to your following blog posts!

Chris, I'm afraid I have to moderate a bit my enthusiasm for what you are doing. I printed your article so I can read it carefully, and something rather unappealing appears right away: What are you trying to do?--Teach a full calculus course (wrongly) while explaining high-level theoretical physics?? You go on and on "explaining" the integral as a "sum"--something that would make any real mathematician cringe (in a basic calculus course we spend weeks explaining limits!); the integral is indeed a linear operator but you really can't call it a "sum"--and then you don't explain the really important derivation. For example (and I would appreciate your explanation for this): On p. 4 (if you can divide this into pages, top of page, how in the world do you go from the definition of v sub I to T=(1/2) Summation...etc. WHERE does the 1/2 come from??? Equally, is it really necessary to explain what the summation symbol means? Or, why in the world would you spend much time deriving the rule for the derivative of a product? I would just state (UV)'=U'V + V'U and be done with it--SPEND your time explaining the DERIVATIONS of the ACTION itself (there are so many holes there, like the one I pointed out above)--not calculus! Assume your readers know calculus, explain the action principle. Thanks.

Hi Amir,

I'm hoping the blog will be fun and useful for everyone with an interest in science, including younger people who might not have had a calculus course yet, and older people who might have missed out on it. So I'll be explaining necessary results from calculus "on the fly", as needed.

You go on and on "explaining" the integral as a "sum"--something that would make any real mathematician cringe (in a basic calculus course we spend weeks explaining limits!)

My aim there was to give a very quick and rough explanation of what an integral is, as part of the explanation of what the action is. Consequences such as the equality of the integral of the rate of change and the net change are obtained later, as needed.

On p. 4 (if you can divide this into pages, top of page, how in the world do you go from the definition of v sub I to T=(1/2) Summation...etc. WHERE does the 1/2 come from???

This formula for the total kinetic energy comes from the definition of as the sum of the kinetic energies of the objects I gave earlier. I didn't directly derive the formula for the kinetic energy of a pointlike object of mass moving at a speed in the post. However the work done on an object by a constant force on it is by definition equal to the force multiplied by the distance the object moves in the direction of the force, and this is equal to the reduction of the potential energy of the object due to that motion. For example, if an object of mass 1 kilogram falls a distance 1 metre near the surface of the Earth, where the force of gravity on it is , then the reduction of its gravitational potential energy is . So from the proof of the conservation of the total energy further down the post, we can work backwards to conclude that the formula for the total kinetic energy, as above, including the factor , is correct.

The reason for explaining the meaning of the summation symbol, , is that I remember from many years ago, when I first encountered it, I had a bit of trouble understanding what it meant. When a young person first learns elementary algebra, they learn that apart from numbers, the arithmetic signs , , , and , and parentheses and , every symbol in a formula, such as , represents a number. The summation symbol does not fit that pattern, but it's very useful, so I thought it would be best to explain what it means.

I added in the proof of the Leibniz rule for the rate of change of a product after first trying to write the rest of the post without it, but it's so useful that in the end I thought it would be best to include it.

Thanks again for the encouragement and your detailed comments, I very much appreciate it.
Thanks, Chris!!
I'll stay tuned to your next installments.
Best of luck,
Amir